NEET Physics Class 12 Chapter 5 Electrostatics MCQ’s

Electrostatics Multiple Choice Question And Answers

Question 1. A body can be negatively charged by

1. Giving excess electrons to it
2. Removing some electrons from it
3. Giving some protons from it
4. Removing some neutrons from it

Solution: 1. Giving excess of electrons to it

Question 2. The minimum charge on an object is

1. 1 coulomb
2. 1 stat coulomb
3. 1.6×10^-19 coulomb
4. 3.2×10^-19 coulomb

Solution: 3. 1.6×10^-19 coulomb

Question 3. A total charge Q is broken into two parts Q1and and they are placed at a distance R from each other. the maximum force of repulsion between them will occur, when

1. \(Q_2=\frac{Q}{R}, Q_1=Q-\frac{Q}{R}\)
2. \(Q_2=\frac{Q}{4}, Q=Q-\frac{2 Q}{3}\)
3. \(Q_2=\frac{Q}{4}, Q_1=\frac{3 Q}{4}\)
4. \(Q_1=\frac{Q}{2}, Q_2=\frac{Q}{2}\)

Solution: 4. \(Q_1=\frac{Q}{2}, Q_2=\frac{Q}{2}\)

Question 4. +2C and +6C two charges are repelling each other with a force of 12N. if each charge is given –2C of charge, the value of the force will be

1. 4N(Attractive)
2. 4N (Repulsive)
3. 8N (Repulsive)
4. Zero

Solution: 4. Zero

Question 5. The magnitude of elective field intensity E is such that an electron placed in it would experience an electrical force equal to its weight given by

1. age
2. \(\frac{\mathrm{mg}}{\mathrm{e}}\)
3. \(\frac{\mathrm{e}}{\mathrm{mg}}\)
4. \(\frac{e^2}{m^2} g\)

Solution: 2. \(\frac{\mathrm{mg}}{\mathrm{e}}\)

Question 6. The distance between the two charges 25μC and 36μC is 11cm At what point on the line joining the two, the intensity will be zero

1. At a distance of 5cm from 25μC
2. At a distance of 5 cm from 36μC
3. At a distance of 10cm from 25μC
4. At a distance of 11cm from 36μC

Solution: 1. At a distance of 5cm from 25μC

Question 7. A charge produces an electric field of 1 N\C at a point distant 0.1 m from it. The magnitude of the charge is

1. 1.11×10^-12C
2. 9.11×10¹²C
3. 7.11×10^-6C
4. None of these

Solution: 1. 1.11×10-12C

Question 8. The angle between the equipotential surface and lines of force is

1. Zero
2. 1800
3. 900
4. 450

Solution: 3. 900

Question 9. A charge of 5C experiences a force of 5000N when it is kept in a uniform electric field. What is the potential difference between two points separated by a distance of 1cm

1. 10 V
2. 250 V
3. 1000 V
4. 2500 V

Solution: 1. 10 V

Question 10. Two equal charges q are placed at a distance of 2a and a third charge –2q is placed at the midpoint, The potential energy of the system is

1. \(\frac{\mathrm{q}^2}{8 \pi \varepsilon_0 \mathrm{a}}\)
2. \(\frac{6 \mathrm{q}^2}{8 \pi \varepsilon_0 \mathrm{a}}\)
3. \(-\frac{7 q^2}{8 \pi \varepsilon_0 a}\)
4. \(\frac{9 q^2}{8 \pi \varepsilon_0 a}\)

Solution: 3. \(-\frac{7 q^2}{8 \pi \varepsilon_0 a}\)

Question 11. A particle of mass ‘m’ and charge ‘q’ is accelerated through a potential difference of V unit, its energy will be

1. qV
2. mqV
3. \(\left(\frac{q}{m}\right) V\)
4. \(\frac{\mathrm{q}}{\mathrm{mV}}\)

Solution: 1. qV

Question 12. When one electron is taken towards the other electron, then the electric potential energy of the system

1. Decreases
2. Increase
3. Remains unchanged
4. Becomes zero

Solution: 2. Increase

Question 13. The electric potential at a point on the axis of an electric dipole depends on the distance r of the point from the dipole as

1. \(\propto \frac{1}{r}\)
2. \(\propto \frac{1}{r^2}\)
3. ∝r
4. \(\propto \frac{1}{r^3}\)

Solution: 2. \(\propto \frac{1}{r^2}\)

Question 14. An electric dipole when placed in a uniform electric field E will have minimum potential energy if the positive direction of dipole moment makes the following angle with E

1. π
2. π/2
3. Zero
4. 3π/2

Solution: 3. Zero

Question 15. An electric dipole of moment P is placed in the position of stable equilibrium in the uniform electric field of intensity E. It is rotated through an angle θ from the initial position. the potential energy of the electric dipole in the final position is :

1. PE cos θ
2. PE sin θ
3. PE (1-cos θ)
4. – PE cos θ

Solution: 4. – PE cos θ

Question 16. The figure shows the electric lines of force emerging from a charged body. If the electric field at A and B are EA and EBrespectively and if the displacement between A and B is r then

1. E_A> E_B
2. E_A < E_B
3. E_A= E_B
4. E_A = E_B

Solution: 1. E_A> E_B

Question 17. The figure shows some of the elective field lines corresponding to an elective field. The figure suggests

1. E_A> E_B> EC
2. E_A= E_B= EC
3. E_A= E_C> E_B
4. E_A= E_C< E_B

Solution: 3. E_A= E_C> E_B

Question 18. A cylinder of radius R and length L is placed in a uniform electric field E parallel to the cylinder axis. The total flux for the surface of the cylinder is given by

1. 2πR²E
2. πR²/E
3. (πR² – πR)E
4. zero

Solution: 4. zero

Question 19. The electric field at a point varies as r0for

1. An electric dipole
2. A point charge
3. A plane infinite sheet of charge
4. A line charge of infinite length

Solution: 3. A plane infinite sheet of charge

Question 20. An electric charge q is placed at the center of a cube of side α. The electric flux on one of its faces will be

1. \(\frac{q}{6 \varepsilon_0}\)
2. \(\frac{\mathrm{q}}{\varepsilon_0 \mathrm{a}^2}\)
3. \(\frac{\mathrm{q}}{4 \pi \varepsilon_0 \mathrm{a}^2}\)
4. \(\frac{\mathrm{q}}{\varepsilon_0}\)

Solution: 1. \(\frac{q}{6 \varepsilon_0}\)

Question 21. The total electric flux coming out of a unit positive charge put in air is

1. ε₀
2. ε₀1
3. (4πε₀)–1
4. 4πε₀ 0

Solution: 2. ε₀₁

Question 22. According to Gauss Theorem electric field of an infinitely long straight wire is proportional to r(3) 31r(4) 1r

1. r
2. \(\frac{1}{r^2}\)
3. \(\frac{1}{r^3}\)
4. \(\frac{1}{r}\)

Solution: 4. \(\frac{1}{r}\)

Question 23. The S.I unit of electric flux is

1. Weber
2. Newton per coulomb
3. Volt ×meter
4. Joule per coulomb

Solution: 3. Volt ×meter

Question 24. A metallic solid sphere is placed in a uniform elective field. The lines of force follow the path (s) shown in the  figure as

1. 1
2. 2
3. 3
4. 4

Solution: 4. 4

Question 25. Inside a hollow charged spherical conductor, the potential

1. Is Constant
2. Varies directly as the distance from the center
3. Varies inversely as the distance from the center
4. Varies inversely as the space of the distance from the center

Solution: 1. Is Constant

Question 26. If q is the charge per unit area on the surface of a conductor, then the electric field intensity at a point on the surface is

1. \(\left(\frac{\mathrm{q}}{\varepsilon_0}\right) \text { normal to surface }\)
2. \(\left(\frac{\mathrm{q}}{2 \varepsilon_0}\right) \text { normal to surface }\)
3. \(\left(\frac{\mathrm{q}}{\varepsilon_0}\right) \text { tangential to surface }\)
4. \(\left(\frac{\mathrm{q}}{2 \varepsilon_0}\right) \text { tangential to surface }\)

Solution: 1. \(\left(\frac{\mathrm{q}}{\varepsilon_0}\right) \text { normal to surface }\)

Question 27. A hollow conducting sphere of radius R has a charge (+Q) on its surface. What is the electric potential within the sphere at a distance r = \(\)From its center

1. Zero (2)
2. \(\frac{1}{4 \pi \varepsilon_0} \frac{Q}{R}\)
3. \(\frac{1}{4 \pi \varepsilon_0} \frac{\mathrm{Q}}{\mathrm{r}^2}\)
4. \(\frac{1}{4 \pi \varepsilon_0} \frac{\mathrm{Q}}{\mathrm{r}}\)

Solution: 3. \(\frac{1}{4 \pi \varepsilon_0} \frac{\mathrm{Q}}{\mathrm{r}^2}\)

Electrostatics Exercise – 1

Section (1)  Properties Of Charge And Coulomb’s Law

Question 1. The relative permittivity of mica is :

1. One
2. Less than one
3. More then one
4. Infinite

Solution: 3. More than one

Question 2. Two identical metallic spheres are charged with 10 and -20 units of charge. If both the spheres are first brought into contact with each other and then are placed in their previous positions, then the ratio of the force in the two situations will be:-

1. -8  1
2. 1: 8
3. -2  1
4. 1: 2

Solution: 1. -8  1

Question 3. Two equal and like charges when placed 5 cm apart experience a repulsive force of 0.144 newtons. The magnitude of the charge in the microcolumn will be :

1. 0.2
2. 2
3. 20
4. 12

Solution: 1. 0.2

Question 4. Two charges of +1 μC and + 5 μC are placed 4 cm apart, the ratio of the force exerted by both charges on each other will be –

1. 1  1
2. 1: 5
3. 5: 1
4. 25: 1

Solution: 1. 1: 1

Question 5. A negative charge is placed at some point on the line joining the two +Q charges at rest. The direction of motion of the negative charge will depend upon the :

1. Position of negative charge alone
2. The magnitude of the negative charge alone
3. Both on the magnitude and position of the negative charge
4. Magnitude of positive charge.

Solution: 1. Position of negative charge alone

Question 6. A body has –80 microcoulomb of charge. Several additional electrons on it will be :

1. 8 x 10^-5
2. 80 x 10¹⁵
3. 5 x 10¹⁴
4. 1.28 x 10^-17

Solution: 3. 5 x 10¹⁴

Question 7. Coulomb’s law for the force between electric charges most closely resembles the following:

1. Law of conservation of energy
2. Newton’s law of gravitation
3. Newton’s 2nd law of motion
4. The law of conservation of charge

Solution: 2. Newton’s law of gravitation

Question 8. A charge Q₁ exerts force on a second charge Q₂. If a 3rd charge Q₃ is brought near, the force of Q₁ is exerted on Q₂.

1. Will increase
2. Will decrease
3. Will remain unchanged
4. Will increase if Q₃ is of the same sign as Q₁ and will decrease if Q₃ is of the opposite sign

Solution: 3. Will remain unchanged

Question 9. A charge particle q₁ is at position (2, – 1, 3). The electrostatic force on another charged particle q₂ at (0, 0, 0) is :

1. \(\\frac{q_1 q_2}{56 \pi \epsilon_0}(2 \hat{i}-\hat{j}+3 \hat{k})\)
2. \(\frac{q_1 q_2}{56 \sqrt{14} \pi \epsilon_0}(2 \hat{i}-\hat{j}+3 \hat{k})\)
3. \(\frac{q_1 q_2}{56 \pi \epsilon_0}(\hat{j}-2 \hat{\mathbf{i}}-3 \hat{k})\)
4. \(\frac{q_1 q_2}{56 \sqrt{14} \pi \epsilon_0}(\hat{j}-2 \hat{i}-3 \hat{k})\)

Solution: 4. \(\frac{q_1 q_2}{56 \sqrt{14} \pi \epsilon_0}(\hat{j}-2 \hat{i}-3 \hat{k})\)

Question 10. Three charges +4q, Q, and q are placed in a straight line of length l at points distance x = 0, x = l/2, and x = respectively. What should the value of Q be to make the net force on q zero?

1. –q
2. –2q
3. –q/2
4. 4q

Solution: 1. –q

Question 11. Two point charges placed at a distance r in the air exert a force F on each other. The value of distance R at which they experience force 4F when placed in a medium of dielectric constant K = 16 is :

1. r
2. r/4
3. r/8
4. 2r

Solution: 3. r/8

Question 12. Two point charges of the same magnitude and opposite sign are fixed at points A and B. A third small point charge is to be balanced at point P by the electrostatic force due to these two charges. The point P:

1. lies on the perpendicular bisector of line AB
2. At the midpoint of line AB
3. Lies to the left of A
4. None of these.

Solution: 4. None of these.

Question 13. A total charge of 20 μC is divided into two parts and placed some distance apart. If the charges experience maximum Colombian repulsion, the charges should be :

1. 5μC, 15 μC
2. 10 μC, 10 μC
3. 12 μC, 8 μC
4. \(\frac{40}{3} \mu C, \frac{20}{3} \mu C\)

Solution: 2. 10 μC, 10 μC

Question 14. Two small spherical balls each carrying a charge Q = 10 μC (10 micro-coulomb) are suspended by two insulating threads of equal lengths 1 each, from a point fixed in the ceiling. It is found that equilibrium threads are separated by an angle of 60º between them, as shown in Fig. What is the tension in the threads (Given \(\frac{1}{\left(4 \pi \varepsilon_0\right)}\)= 9 × 10⁹ Nm/C²) – 0

1. 18 N
2. 1.8 N
3. 0.18 N
4. None of the above

Solution: 2. 1.8 N

Question 15. The separation between the two charges +q and – q becomes double. The value of force will be

1. Twofoldld
2. HalFour
3. folded
4. One fourth

Solution: 4. One fourth

Question 16. The dielectric constant K of an insulator can be :

1. 5
2. 0.5
3. –1
4. zero

Solution: 1. 5

Question 17. Two spherical conductors B and C having equal radii and carrying equal charges in them repel each other with a force F when kept apart at some distance. A third spherical conductor having the same radius as that of B but uncharged is brought in contact with B, then brought in contact with C, and finally removed away from both. The new force of repulsion between B and C is

1. \(\frac{F}{4}\)
2. \(\frac{3 F}{4}\)
3. \(\frac{F}{8}\)
4. \(\frac{3 F}{8}\)

Solution: 4. \(\frac{3 F}{8}\)

Question 18. Three charges –q₁, + q_2, and –are placed as shown in the figure. The x-component of the force on –q₁ is proportional to :

1. \(\frac{q_2}{b^2}-\frac{q_3}{a^2} \cos \theta\)
2. \(\frac{q_2}{b^2}+\frac{q_3}{a^2} \sin \theta\)
3. \(\frac{q_2}{b^2}+\frac{q_3}{a^2} \cos \theta\)
4. \(\frac{q_2}{b^2}-\frac{q_3}{a^2} \sin \theta\)

Solution: 2. \(\frac{q_2}{b^2}+\frac{q_3}{a^2} \sin \theta\)

Question 19. Two spherical conductors B and C having equal radii and carrying equal charges repel each other with a force F when kept apart at some distance. A third spherical conductor having the same radius as that of B but uncharged is brought in contact with B, then brought in contact with C, and finally removed away from both. The new force of repulsion between B and C is :

1. \(\frac{F}{4}\)
2. \(\frac{3 F}{4}\)
3. \(\frac{F}{8}\)
4. \(\frac{3 F}{8}\)

Solution: 4. \(\frac{3 F}{8}\)

Question 20. Under the influence of the Coulomb field of charge +Q, a charge –q is moving around it in an elliptical orbit. Find out the correct statement(s).

1. The angular momentum of the charge –q is constant
2. The linear momentum of the charge –q is constant
3. The angular velocity of the charge – q is constant
4. The linear speed of the charge –q is constant

Solution: 1. The angular momentum of the charge –q is constant

Question 21. When the charge is given to a soap bubble, it shows :

1. An increase in size `
2. Sometimes an increase and sometimes a decrease in size
3. No change in size
4. None of these

Solution: 1. An increase in size `

Section (2): Electric Field

Question 1. If an electron is placed in a uniform electric field, then the electron will :

1. Experience no force.
2. Moving with constant velocity in the direction of the field.
3. Move with constant velocity in the direction opposite to the field.
4. Accelerate in a direction opposite to the field.

Solution: 4. Accelerate in a direction opposite to the field.

Question 2. If Q = 2 column and force on it is F = 100 newton, then the value of field intensity will be :

1. 100 N/C
2. 50 N/C
3. 200 N/C
4. 10 N/C

Solution: 2. 200 N/C

Question 3. Two infinite linear charges are placed parallel at 0.1 m apart. If each has a charge density of 5μ C/m, then the force per unit length of one of the linear charges in N/m is :

1. 2.5
2. 3.25
3. 4.5
4. 7.5

Solution: 3. 4.5

Question 4. The electric field intensity due to a uniformly charged sphere is zero :

1. At the center
2. At infinity
3. At the center and an infinite distance
4. On the surface

Solution: 3. At the center and an infinite distance

Question 5. Two spheres of radii 2 cm and 4 cm are charged equally, then the ratio of charge density on the surfaces of the spheres will be –

1. 1: 2
2. 4: 1
3. 8: 1
4. 1: 4

Solution: 2. 4: 1

Question 6. The total charge on a sphere of radii 10 cm is 1 μC. The maximum electric field due to the sphere in N/C will be –

1. 9 x 10^-5
2. 9 x 10³
3. 9 x 10⁵
4. 9 x 10¹⁵

Solution: 3. 9 x 10⁵

Question 7. A charged water drop of radius 0.1 μm is under equilibrium in some electric field. The charge on the drop is equivalent to an electronic charge. The intensity of the electric field is (g = 10 m/s²)-

1. 1.61 NC^-1
2. 26.2 NC^-1
3. 262 NC^-1
4. 1610 NC^-1

Solution: 3. 262 NC^-1

Question 8. Two large-sized charged plates have a charge density of +σ and -σ. The resultant force on the proton located midway between them will be –

1. σ ∈ e/ 0
2. σ e / 2 ∈ 0
3. 2 e/ σ ∈ 0
4. zero

Solution: 1. σ ∈ e/ 0

Question 9. Two parallel charged plates have a charge density of +σ and -σ. The resultant force on the proton located outside the plates at some distance will be –

1. 2e/ σ∈0
2. σe/∈0
3. σe / 2 ∈ 0
4. zero

Solution: 4. zero

Question 10. The charge density of an insulating infinite surface is (e/π) C/m² then the field intensity at a nearby point in volt/meter will be –

1. 2.88 x 10^-12
2. 2.88 x 10^-10
3. 2.88 x 10^-9
4. 2.88 x 10^-19

Solution: 3. 2.88 x 10^-9

Question 11. There is a uniform electric field in the x-direction. If the work done by an external agent in moving a charge of 0.2 C through a distance of 2 meters slowly along the line making an angle of 60º with x-direction is 4 joule, then the magnitude of E is :

1. 3 N / C
2. 4 N/C
3. 5 N/C
4. 20 N/C

Solution: 4. 20 N/C

Question 12. A simple pendulum has a length of L and a mass of bob m. The bob is given a charge q coulomb. The pendulum is suspended in a uniform horizontal electric field of strength E as shown in the figure, then calculate the period of oscillation when the bob is slightly displaced from its mean position is: E

1. \(2 \pi \sqrt{\frac{\ell}{g}}\)
2. \(2 \pi \sqrt{\left\{\frac{\ell}{g+\frac{q E}{m}}\right\}}\)
3. \(2 \pi \sqrt{\left\{\frac{\ell}{g-\frac{q E}{m}}\right\}}\)
4. \(2 \pi \sqrt{\frac{\ell}{\sqrt{g^2+\left(\frac{q E}{m}\right)^2}}}\)

Solution: 4. \(2 \pi \sqrt{\frac{\ell}{\sqrt{g^2+\left(\frac{q E}{m}\right)^2}}}\)

Question 13. Charge 2Q and –Q are placed as shown in the figure. The point at which electric field intensity is zero will be:

1. Somewhere between –Q and 2Q
2. Somewhere on the left of –Q
3. Somewhere on the right of 2Q
4. Somewhere on the right bisector of line joining –Q and 2Q

Solution: 2. Somewhere on the left of –Q

Question 14. The maximum electric field intensity on the axis of a uniformly charged ring of charge q and radius R will be :

1. \(\frac{1}{4 \pi \varepsilon_0} \frac{q}{3 \sqrt{3} R^2}\)
2. \(\frac{1}{4 \pi \varepsilon_0} \frac{2 q}{3 R^2}\)
3. \(\frac{1}{4 \pi \varepsilon_0} \frac{2 q}{3 \sqrt{3} R^2}\)
4. \(\frac{1}{4 \pi \varepsilon_0} \quad \frac{3 q}{2 \sqrt{3} R^2}\)

Solution: 3. \(\frac{1}{4 \pi \varepsilon_0} \frac{2 q}{3 \sqrt{3} R^2}\)

Question 15. A charged particle of charge q and mass m is released from rest in a uniform electric field E. Neglecting the effect of gravity, the kinetic energy of the charged particle after time ‘t’ seconds is

1. \(\frac{\text { Eqm }}{\mathrm{t}}\)
2. \(\frac{E^2 q^2 t^2}{2 m}\)
3. \(\frac{2 E^2 t^2}{m q}\)
4. \(\frac{E q^2 m}{2 t^2}\)

Solution: 2. \(\frac{E^2 q^2 t^2}{2 m}\)

Question 16. The electric field above a uniformly charged nonconducting sheet is E. If the nonconducting sheet is now replaced by a conducting sheet, with the charge same as before, the new electric field at the same point is :

1. 2E
2. E
3. E/2
4. None of these

Solution: 2. E

Question 17. The linear charge density on the upper half of a segment of a ring is λ and at the lower half, it is – λ. The direction of the electric field at the center O of the ring is :

1. along OA
2. along OB
3. along OC
4. along OD

Solution: 3. along OC

Question 18. The given figure gives electric lines of force due to two charges q₁ and q₂. What are the signs of the two charges?

1. imageBoth are negative
2. Both are positive
3. q₁is positive but q₂is negative
4. q₁is negative but q₂is positive

Solution: 1. Both are negative

Question 19. A positively charged pendulum is oscillating in a uniform electric field as shown in Figure. Its period of SHM is compared to that when it was uncharged. (mg > qE)

1. Will increase
2. Will decrease
3. Will not change
4. Will first increase then decrease

Solution: 1. Will increase

Question 20. A +q₁charge is at the center of an imaginary spherical Gaussian surface ‘S’, and a – q₁ charge is placed near this +q₁charge inside ‘S’. A charge +q₂is located outside this Gaussian surface. Then electric field on the Gaussian surface will be

1. Due to – q₁and q₂
2. Uniform
3. Due to all charges
4. Zero

Solution: 3. Due to all charges

Question 21. Three large parallel plates have uniform surface charge densities as shown in the figure. Find out the electric field intensity at point P.

1. \(-\frac{4 \sigma}{\epsilon_0} \hat{\mathrm{k}}\)
2. \(\frac{4 \sigma}{\epsilon_0} \hat{k}\)
3. \(-\frac{2 \sigma}{\epsilon_0} \hat{\mathrm{k}}\)
4. \(\frac{2 \sigma}{\epsilon_0} \hat{k}\)

Solution: 3. \(-\frac{2 \sigma}{\epsilon_0} \hat{\mathrm{k}}\)

Question 22. The wrong statement about electric lines of force is –

1. These originate from positive charge and end on negative charge
2. They do not intersect each other at a point
3. They have the same form for a point charge and a sphere(outside the sphere)
4. They have physical existences

Solution: 4. They have physical existences

Question 23. The insulation property of air breaks down at intensity as 3 × 10⁶ V/m. The maximum charge that can be given to a sphere of diameter 5 m is :

1. 2 × 10^-2 C
2. 2 × 10^-3 C
3. 2 × 10^-4 C
4. 0

Solution: 2. 2 × 10^-3 C

Question 24. Choose the correct statement regarding electric lines of force :

1. Emerges from (–υe) charge and meet from (+υe) charge
2. Where the electric lines of force are close electric field in that region is strong
3. Just as it is shown for a point system in the same way it represents for a solid sphere
4. Has a physical nature

Solution: 2. Where the electric lines of force are close electric field in that region is strong

Question 25. The electric field required to keep a water drop of mass m and charge e just to remain suspended is :

1. mg
2. EMG
3. mg/e
4. em/g

Solution: 3. mg/e

Question 26. Two parallel large thin metal sheets have equal surface charge densities (σ = 26.4 × 10^-12 C/m2) of opposite signs. The electric field between these sheets is

1. 1.5 N/C
2. 1.5 × 10^-10 N/C
3. 3 N/C
4. 3 × 10^-10 N/C

Solution: 3. 3 N/C

Question 27. A charged ball B hangs from a silk thread S, which makes an angle θ with a large charged conducting sheet P, as shown in the figure. The surface charge density σ of the sheet is proportional to

1. cot θ
2. cos θ
3. tan θ
4. sin θ

Solution: 3. tan θ

Question 28. The electric potential at a point in free space due to a charge Q coulomb is Q × 10¹¹ V. The electric field at that point is

1. 4π ε₀ Q × 10²² V/m
2. 12π ε₀ Q × 10²⁰ V/m
3. 4π ε₀ Q × 10²⁰ V/m
4. 12π ε₀ Q × 10²² V/m

Solution: 1. 4π ε₀ Q × 10²² V/m

Question 29. A thin conducting ring of radius R is given a charge +Q. The electric field at the center O of the ring due to the charge on the part AKB of the ring is E. The electric field at the center due to the charge on the part ACDB of the ring is

1. 3E along KO
2. E along OK
3. E along KO
4. 3 E along OK

Solution: 2. E along OK

Question 30. A charged ball B hangs from a silk thread S, which makes an angle θ with a large charged conducting sheet P, as shown in the figure. The surface charge density σ of the sheet is proportional to:

1. sin θ
2. tanθ
3. cosθ
4. cot θ

Solution: 2. tanθ

Question 31. Two point charges + 8 q and – 2q are located at x = 0 and x = L respectively. The location of a point on the x-axis at which the net electric field due to these two point charges is zero is:

1. 8L
2. 4L
3. 2L
4. L/4

Solution: 3. 2L

Question 32. Two spherical conductors A and B of radii 1 mm and 2mm are separated by a distance of 5 cm and are uniformly charged. If the spheres are connected by a conducting wire then in equilibrium condition, the ratio of the magnitude of the electric fields at the surfaces of spheres A and B is :

1. 2: 1
2. 1: 4
3. 4: 1
4. 1: 2

Solution: 1. 2: 1

Question 33. A thin spherical shell of radius R has charge Q spread uniformly over its surface. Which of the following graphs most closely represents the electric field E (r) produced by the shell in the range 0 < r < ∞, where r is the distance from the center of the shell?

Solution: 4.

Question 34. The figure shows the electric lines of force emerging from a charged body. If the electric fields at A and B are and respectively and if the distance between A and B is r, then

1. E_A< E_B
2. E_A> E_B
3. \(E_A=\frac{E_B}{r}\)
4. \(E_A=\frac{E_B}{r^2}\)

Solution: 2. E_A> E_B

Question 35. Two point charges a and b, whose magnitudes are the same are positioned at a certain distance from each other with a at the origin. The graph is drawn between electric field strength at points between a and b and distance x from a. E is taken positive if it is along the line joining from a to b. From the graph, it can be decided that

1. A is positive, B is negative
2. A and B both are positive
3. A and B both are negative
4. A is negative, B is positive

Solution: 1. A is positive, B is negative

Question 36. A wooden block performs SHM on a frictionless surface with frequency, ν0. The block carries a charge +Q on its surface. If now a uniform electric field E is switched on as shown, then the SHM of the block will be

1. Of the same frequency and with a shifted mean position.
2. Of the same frequency and with the same mean position.
3. Of changed frequency and with shifted mean position.
4. Of changed frequency and with the same mean position.

Solution: 1. Of the same frequency and with shifted mean position.

Question 37. A charged oil drop is suspended in a uniform field of 3 × 10⁴ V/m so that it neither falls nor rises. The charge on the drop will be (Take the mass of the drop = 9.9 × 10^-15 kg and g = 10 m/s2)

1. 3.3 × 10^-18C
2. 3.2 × 10^-18 C
3. 1.6 × 10^-18 C
4. 4.8 × 10^-18 C

Solution: 1. 3.3 × 10^-18 C

Question 38. Three positive charges of equal value q are placed at the vertices of an equilateral triangle. The resulting lines of force should be sketched as in :

Solution: 2.

Section (3): Electric Potential And Potential Difference

Question 1. If we move in a direction opposite to the electric lines of force :

1. Electrical potential decreases.
2. Electrical potential increases.
3. Electrical potential remains uncharged
4. Nothing can be said.

Solution: 2. Electrical potential increases.

Question 2. The distance between two plates is 2 cm when an electric potential of 10 volts is applied to both plates, then the value of the electric field will be –

1. 20 N/C
2. 500 N/C
3. 5 N/C
4. 250 N/C

Solution: 2. 500 N/C

Question 3. Two objects A and B are charged with equal charge Q. The potential of A relative to B will be –

1. More
2. Equal
3. Less
4. Indefinite

Solution: 4. Indefinite

Question 4. In electrostatics the potential is equivalent to –

1. Temperature in heat
2. Height of levels in liquids
3. Pressure in gases
4. All of the above

Solution: 4. All of the above

Question 5. The potential due to a point charge at distance r is –

1. Proportional to r.
2. Inversely proportional to r.
3. Proportional to r₂.
4. Inversely proportional to r₂

Solution: 2. Inversely proportional to r.

Question 6. The dimensions of potential difference are –

1. ML²T–2Q^-1
2. MLT–2Q^-1
3. MT^-2Q^-2
4. ML2T^-2Q^-1

Solution: 1. ML2T^-2Q–1

Question 7. An object is charged with a positive charge. The potential at that object will be –

1. Positive only
2. Negative only
3. Zero always
4. May be positive, negative, or zero.

Solution: 4. May be positive, negative or zero.

Question 8. Two points (0, a) and (0, -a) have charges q and -q respectively then the electrical potential at the origin will be

1. Zero
2. Q/a
3. Q/2a
4. Q/4a₂

Solution: 1. Zero

Question 9. The charges of the same magnitude q are placed at four corners of a square of side a. The value of the potential at the center of the square will be –

1. 4kq/a
2. 4 √2kq /a
3. 4kq √2a
4. kg / a √2

Solution: 2. 4 V2kq /a

Question 10. Three equal charges are placed at the three corners of an isosceles triangle as shown in the figure. The statement which is true for electric potential V and the field intensity E at the center of the triangle –

1. V = 0, E = 0
2. V = 0, E ≠ 0
3. V ≠ 0, E = 0
4. V ≠ 0, E ≠ 0

Solution: 3. V ≠ 0, E = 0

Question 11. The potential at 0.5 Å from a proton is –

1. 0.5 volt
2. 8μ volt
3. 28.8 volt
4. 2 volt

Solution: 3. 28.8 volt

Question 12. A wire of 5 m in length carries a steady current. If it has an electric field of 0.2 V/m, the potential difference across the wire in volts will be –

1. 25
2. 0.04
3. 1.0
4. None of the above

Solution: 3. 1.0

Question 13. An infinite number of charges of equal magnitude q, but alternate charges of opposite sign are placed along the x-axis at x = 1, x = 2, x = 4, x =8,… and so on. The electric potential at the point x = 0 due to all these charges will be –

1. kq/2
2. kq/3
3. 2kq/3
4. 3kq/2

Solution: 3. 2kq/3

Question 14. The electric potential inside a uniformly positively charged non-conducting solid sphere has the value which –

1. Increase with increases in distance from the center.
2. Decreases with increases in distance from the center.
3. Is equal at all the points.
4. Is zero at all the points.

Solution: 2. Decreases with increases in distance from the center.

Question 15. Two metallic spheres which have equal charges, but their radii are different, are made to touch each other and then separated apart. The potential spheres will be –

1. Same as before
2. More for bigger
3. More for smaller
4. Equal

Solution: 4. Equal

Question 16. Two spheres of radii R and 2R are given a source equally positively charged and then connected by a long conducting wire, then the positive charge will

1. Flow from the smaller sphere to the bigger sphere
2. Flow from the bigger sphere to the smaller sphere
3. Not flow.
4. Oscillate between the spheres.

Solution: 1. Flow from the smaller sphere to the bigger sphere

Question 17. The potential difference between two isolated spheres of radii r₁ and r₂ is zero. The ratio of their charges Q₁/Q₂ will be

1. r₁/r₂
2. r₂/r₁
3. r₁₂/r₂₂
4. r₁₃/r₂₃

Solution: 1. r1/r₂

Question 18. The potential on the conducting spheres of radii r₁ and r₂ is the same, the ratio of their charge densities will be

1. r₁/r₂
2. r₂/r₁
3. r₁₂/r₂₂
4. r₂₂/r₁₂

Solution: 2. r₂/r1

Question 19. 64 charged drops coalesce to form a bigger charged drop. The potential of the bigger drop will be times that of a smaller drop –

1. 4
2. 16
3. 64
4. 8

Solution: 2. 16

Question 20. The electric potential outside a uniformly charged sphere at a distance ‘r’ is (‘a’ being the radius of the sphere)-

1. Directly proportional to a3
2. Directly proportional to r.
3. Inversely proportional to r.
4. Inversely proportional to a3.

Solution: 3. Inversely proportional to r.

Question 21. A conducting shell of radius 10 cm is charged with 3.2 x 10^-19 C. The electric potential at a distance of 4cm from its center in volt be –

1. 9 x 10^-9
2. 288
3. 2.88 x 10^-8
4. Zero

Solution: 3. 2.88 x 10^-8

Question 22. At a certain distance from a point charge the electric field is 500 V/m and the potential is 3000 V. What is the distance?

1. 6 m
2. 12 m
3. 36 m
4. 144 m

Solution: 1. 6 m

Question 23. The figure represents a square carrying charges +q, +q, –q, –q at its four corners as shown. Then the potential will be zero at points

1. A, B, C, P, and Q
2. A, B, and C
3. A, P, C, and Q
4. P, B, and Q

Solution: 2. A, B, and C

Question 24. Two equal positive charges are kept at points A and B. The electric potential at the points between A and B (excluding these points) is studied while moving from A to B. The potential

1. Continuously increases
2. Continuously decreases
3. Increases then decreases
4. Decreases than increases

Solution: 4. Decreases than increases

Question 25. A semicircular ring of radius 0.5 m is uniformly charged with a total charge of 1.5 × 10–9 coul. The electric potential at the center of this ring is :

1. 27 V
2. 13.5 V
3. 54 V
4. 45.5 V

Solution: 1. 27 V

Question 26. The kinetic energy that an electron acquires when accelerated (from rest) through a potential difference of 1 volt is called :

1. 1 joule
2. 1 electron volt
3. 1 erg
4. 1 watt

Solution: 2. 1 electron volt

Question 27. The potential difference between points A and B in the given uniform electric field is :

1. Ea
2. \(E \sqrt{\left(a^2+b^2\right)}\)
3. Eb
4. (Eb/√2)

Solution: 3. Eb

Question 28. A particle of charge Q and mass m travels through a potential difference V from rest. The final momentum of the particle is :

1. \(\frac{\mathrm{mV}}{\mathrm{Q}}\)
2. \(2 Q \sqrt{m V}\)
3. \(\sqrt{2 m Q V}\)
4. \(\sqrt{\frac{2 Q V}{m}}\)

Solution: 3. \(\sqrt{2 m Q V}\)

Question 29. If a uniformly charged spherical shell of radius 10 cm has a potential V at a point distant 5 cm from its center, then the potential at a point distant 15 cm from the center will be :

1. \(\frac{V}{3}\)
2. \(\frac{2 V}{3}\)
3. \(\frac{3}{2} \mathrm{~V}\)
4. 3V

Solution: 2. \(\frac{2 V}{3}\)

Question 30. A hollow conducting sphere of radius R has a charge (+Q) on its surface. What is the electric potential within the sphere at a distance r =\(\frac{R}{3}\)from its center

1. zero
2. \(\frac{1}{4 \pi \varepsilon_0} \frac{\mathrm{Q}}{\mathrm{r}}\)
3. \(\frac{1}{4 \pi \varepsilon_0} \frac{Q}{R}\)
4. \(\frac{1}{4 \pi \varepsilon_0} \frac{\mathrm{Q}}{\mathrm{r}^2}\)

Solution: 3. \(\frac{1}{4 \pi \varepsilon_0} \frac{Q}{R}\)

Question 31. Consider a thin spherical shell of radius R with its center at the origin, carrying uniform positive surface charge density. The variation of the magnitude of the electric field \(|\vec{E}(r)|\)and the electric potential V(r) with the distance r from the center, is best represented by which graph?

Solution: 4.

Question 32. The electric field at a point 20 cm away from the center of the dielectric sphere is 100 V/m, the radius of the sphere is 10 cm, then the value of the electric field at a distance 3 cm from the center is :

1. 100 V/m
2. 125 V/m
3. 120 V/m
4. 0

Solution: 4. 0

Question 33. If n drops of potential V merge, find new potential on the big drop :

1. n2/3 V
2. n1/3 V
3. nV
4. Vn/3

Solution: 1. n2/3 V

Question 34. Two conducting spheres of radii R1 and respectively are charged and joined by a wire. The ratio of electric fields of spheres is

1. \(\frac{\mathrm{R}_2^2}{\mathrm{R}_1^2}\)
2. \(\frac{R_1^2}{R_2^2}\)
3. \(\frac{R_2}{R_1}\)
4. \(\frac{R_1}{R_2}\)

Solution: 3. \(\frac{R_2}{R_1}\)

Question 35. Charge on a sphere of radius R is q and on the sphere of radius 2R is –2q. If these spheres are connected through a conducting wire then, the amount of charge flown through the wire will be :

1. \(-\frac{q}{3}\)
2. \(\frac{2 q}{3}\)
3. q
4. \(\frac{4 q}{3}\)

Solution: 4. \(\frac{4 q}{3}\)

Question 36. Two identical conducting spheres R and S have negative charges Q1and Q2respectively, but Q1 Q2. The spheres are brought to touch each other and then kept in their original positions, now the force between them is

1. Greater than that before the spheres touched
2. Less than that before the spheres touched
3. Same as before the spheres
4. Zero

Solution: 1. Greater than that before the spheres touched

Question 37. 27 smaller drops combine to form a bigger drop if the potential on a smaller drop is v then the potential on a bigger drop will be

1. 9V
2. 3V
3. 27V
4. 1/3V

Solution: 1. 9V

Question 38. A thin spherical conducting shell of radius R has a charge q. Another charge Q is placed at the center of the shell. The electrostatic potential at a point P at a distance R/2 from the center of the shell is :

1. \(\frac{2 \mathrm{Q}}{4 \pi \varepsilon_0 \mathrm{R}}\)
2. \(\frac{2 Q}{4 \pi \varepsilon_0 R}-\frac{2 q}{4 \pi \varepsilon_0 R}\)
3. \(\frac{2 \mathrm{Q}}{4 \pi \varepsilon_0 \mathrm{R}}+\frac{\mathrm{q}}{4 \pi \varepsilon_0 \mathrm{R}}\)
4. \(\frac{(q+Q)}{4 \pi \varepsilon_0 R} \frac{2}{R}\)

Solution: 3. \(\frac{2 \mathrm{Q}}{4 \pi \varepsilon_0 \mathrm{R}}+\frac{\mathrm{q}}{4 \pi \varepsilon_0 \mathrm{R}}\)

Question 39. A charged oil drop is suspended in a uniform field of 3 × 104 V/m so that it neither falls nor rises. The charge on the drop will be : (take the mass of the charge = 9.9 × 10^-15 kg, g = 10m/sec2)

1. 3.3 × 10^-18 C
2. 3.2 × 10^-18 C
3. 1.6 × 10^-18 C
4. 4.8 × 10^-18 C

Solution: 1. 3.3 × 10^-18 C

Question 40. Two thin wire rings, each having a radius R are placed at a distance d apart with their axes coinciding. The charges on the two rings are + q and –q. The potential difference between the centers of the two rings is:

1. zero
2. \(\frac{q}{4 \pi \varepsilon_0}\left[\frac{1}{R}-\frac{1}{\sqrt{R^2+d^2}}\right]\)
3. \(\frac{q R}{4 \pi \varepsilon_0 d^2}\)
4. \(\frac{q}{2 \pi \varepsilon_0}\left[\frac{1}{R}-\frac{1}{\sqrt{R^2+d^2}}\right]\)

Solution: 4. \(\frac{q}{2 \pi \varepsilon_0}\left[\frac{1}{R}-\frac{1}{\sqrt{R^2+d^2}}\right]\)

Question 41. An electric charge of 10–3µC is placed at the origin (0, 0) of the X–Y coordinate system. Two points A and B are situated at will be (V₂, V₂ ) and (2, 0) respectively. The potential difference between the points A and B

1. 9 volt
2. zero
3. 2 volt
4. 4.5 volt

Solution: 2. zero

Question 42. Charges are placed on the vertices of a square as shown. Let Ebe be the electric field and V the potential at the center. If the charges on A and B are interchanged with those on D and C respectively, then

1. \(\text { E }\) remains unchanged, V changes
2. Both\(\text { E }\) and V change
3. \(\text { E }\) and V remain unchanged
4. \(\text { E }\) changes, V remains unchanged

Solution: 4. \(\text { E }\) changes, V remains unchanged

Question 43. A hollow uniformly charged sphere has a radius of r. If the potential difference between its surface and a point at a distance 3r from the center is V, then the electric field intensity at a distance 3r from the center is:

1. V/6r
2. V/4r
3. V/3r
4. V/2r

Solution: 1. V/6r

Question 44. A hollow sphere of radius 5 cm is uniformly charged such that the potential on its surface is 10 volts then the potential at the center of the sphere will be :

1. Zero
2. 10 volt
3. Same as at a point 5 cm away from the surface
4. Same as at a point 25 cm away from the center

Solution: 2. 10 volt

Section (4): Electric Potential Energy Of A Particle

Question 1. A nucleus has a charge of + 50e. A proton is located at a distance of 10^-12 m. The potential at this point in volt will be –

1. 14.4 x 10⁴
2. 7.2 x 10⁴
3. 7.2 x 10^-12
4. 14.4 x 10⁸

Solution: 2. 7.2 x 10⁴

Question 2. Under the influence of charge, a point charge q is carried along different paths from point A to point B, then work done will be –

1. Maximum for path four.
2. Maximum for path one.
3. Equal for all paths
4. Minimum for path three.

Solution: 3. Equal for all paths

Question 3. An electron moving in an electric potential field V1enters a higher electric potential field V2, then the change in kinetic energy of the electron is proportional to –

1. (V₂ — V₁)1/2
2. V₂ — V₁
3. (V₂ — V₁)₂
4. \(\frac{\left(V_2-V_1\right)}{V_2}\)

Solution: 2. V₂— V₁

Question 4. In the electric field of charge Q, another charge is carried from A to B. A to C, A to D, and A to E, then work done will be –

1. Minimum along path AB.
2. Minimum along path AD.
3. Minimum along path AE.
4. Zero along all the paths.

Solution: 4. Zero along all the paths.

Question 5. The work done to take an electron from rest where the potential is – 60 volts to another point where the potential is – 20 volts is given by –

1. 40 eV
2. –40 eV
3. 60 eV
4. –60 eV

Solution: 2. –40 eV

Question 6. If a charge is shifted from a low-potential region to high high-potential region. the electrical potential energy:

1. Increases
2. Decreases
3. Remains constant
4. May increase or decrease.

Solution: 4. May increase or decrease.

Question 7. A particle A has a charge +q and particle B has a charge + 4q with each of them having the same mass m. When allowed to fall from rest through the same electrical potential difference, the ratio of their speed vA: vB will be :

1. 2: 1
2. 1: 2
3. 4: 1
4. 1: 4

Solution: 2. 1: 2

Question 8. In an electron gun, electrons are accelerated through a potential difference of V volt. Taking electronic charge and mass to be respectively e and m, the maximum velocity attained by them is :

1. \({\frac{2 \mathrm{eV}}{\mathrm{m}}}\)
2. \(\sqrt{\frac{2 \mathrm{eV}}{\mathrm{m}}}\)
3. 2 m/eV
4. (√2/2em)

Solution: 2. \(\sqrt{\frac{2 \mathrm{eV}}{\mathrm{m}}}\)

Question 9. In a cathode ray tube, if V is the potential difference between the cathode and anode, the speed of the electrons, when they reach the anode is proportional to : (Assume initial velocity = 0)

1. V
2. 1/√-5
3. √V
4. √2

Solution: 3. √V

Question 10. An electron of mass m and charge e is accelerated from rest through a potential difference V in a vacuum. The final speed of the electron will be –

1. \(\mathrm{V} \sqrt{\mathrm{e} / \mathrm{m}}\)
2. \(\sqrt{\mathrm{eV} / \mathrm{m}}\)
3. \(\sqrt{2 \mathrm{eV} / \mathrm{m}}\)
4. \(2 \mathrm{eV} / \mathrm{m}\)

Solution: 3. \(\sqrt{2 \mathrm{eV} / \mathrm{m}}\)

Question 11. Positive and negative point charges of equal magnitude are kept \(\left(0,0, \frac{a}{2}\right)\)and \(\left(0,0, \frac{-a}{2}\right)\) respectively. The work done by the electric field when another positive point charge is moved from (–a, 0, 0) to (0, a, 0) is

1. Positive
2. Negative
3. Zero
4. Depends on the path connecting the initial and final positions.

Solution: 3. Zero

Question 12. If a positive charge is shifted from a low potential region to a high potential region, then electric potential energy

1. Decreases
2. Increases
3. Remains the same
4. May increase or decrease

Solution: 2. Increases

Question 13. An electron is accelerated by 1000 volts, potential difference, and its final velocity is :

1. 3.8 × 10⁷ m/s
2. 1.9 × 10⁶ m/s
3. 1.9 × 10⁷ m/s
4. 5.7 × 10⁷ m/s

Solution: 3. 1.9 × 10⁷ m/s

Question 14. As per this diagram, a point charge +q is placed at the origin O. Work done in taking another point charge –Q from the point A [co-ordiantes (o, a)] to another point B [co-ordinates(a,o)] along the straight path AB is :

1. Zero
2. \(\left(\frac{-q Q}{4 \pi \varepsilon_0} \frac{1}{\mathrm{a}^2}\right) \sqrt{2 \mathrm{a}}\)
3. \(\left(\frac{\mathrm{qQ}}{4 \pi \varepsilon_0} \frac{1}{\mathrm{a}^2}\right) \cdot \frac{\mathrm{a}}{\sqrt{2}}\)
4. \(\left(\frac{\mathrm{qQ}}{4 \pi \varepsilon_0} \frac{1}{\mathrm{a}^2}\right) \sqrt{2 a}\)

Solution: 1. Zero

Question 15. Two charges q₁ and q₂ are placed 30 cm apart, as shown in the figure. A third charge q3is moved along the arc of a circle of radius 40 cm from C to D. The change in the potential energy of the system is \(\frac{\mathrm{q}_3}{4 \pi \varepsilon_0}\)k, where k is :

1. 8q₂
2. 8q₁
3. 6q₂
4. 6q₁

Solution: 1. 8q₂

Question 16. Charges +q and –q are placed at points A and B respectively which are a distance 2 L apart, C is the midpoint between A and B. The work done in moving a charge +Q along the semicircle CRD is :

1. \(\frac{\mathrm{qQ}}{4 \pi \varepsilon_0 \mathrm{~L}}\)
2. \(\frac{\mathrm{qQ}}{2 \pi \varepsilon_0 \mathrm{~L}}\)
3. \(\frac{\mathrm{qQ}}{6 \pi \varepsilon_0 \mathrm{~L}}\)
4. \(-\frac{\mathrm{qQ}}{6 \pi \varepsilon_0 \mathrm{~L}}\)

Solution: 4. \(-\frac{\mathrm{qQ}}{6 \pi \varepsilon_0 \mathrm{~L}}\)

Question 17. A charged particle ‘q’ is shot towards another charged particle ‘Q’, which is fixed, with a speed ‘v’. It approaches ‘Q’ up to the closest distance r and then returns. If q were given a speed of ‘2v’, the closest distance of approach would be :

1. r
2. 2r
3. r²
4. r⁴

Solution: 4. r⁴

Question 18. Two insulating plates are both uniformly charged in such a way that the potential difference between them is V₂– V₁= 20 V. (i.e. plate 2 is at a higher potential). The plates are separated by d = 0.1 m and can be treated as infinitely large. An electron is released from rest on the inner surface of plate 1. What is its speed when it hits plate 2? (e = 1.6 × 10^-19 C, me= 9.11 × 10^-31 kg)

1. 1.87 × 10⁶ m/s
2. 32 × 10^-19m/s
3. 2.65 × 10⁶ m/s
4. 7.02 × 10¹² m/s

Solution: 3. 2.65 × 10⁶ m/s

Question 19. A particle of mass 2 g and charge 1μC is held at rest on a frictionless horizontal surface at a distance of 1 m from a fixed charge of 1 mC. If the particle is released it will be repelled. The speed of the particle when it is at a distance of 10 m from the fixed charge is:

1. 100 m/s
2. 90 m/s
3. 60 m/s
4. 45 m/s

Solution: 2. 90 m/s

Question 20. On moving a charge of 20 coulombs by 2 cm, 2 J of work is done, then the potential difference between the points is :

1. 0.1 V
2. 8 V
3. 2 V
4. 0.5 V

Solution: 1.0.1 V

Question 21. For an infinite line of charge having charge density λ lying along the x-axis, the work required in moving charge q from C to A along arc CA is :

1. \(\frac{\mathrm{q} \lambda}{\pi \varepsilon_0} \log _{\mathrm{e}} \sqrt{2}\)
2. \(\frac{\mathrm{q} \lambda}{4 \pi \varepsilon_0} \log _e \sqrt{2}\)
3. \(\frac{\mathrm{q} \lambda}{4 \pi \varepsilon_0} \log _{\mathrm{e}} 2\)
4. \(\frac{q \lambda}{2 \pi \varepsilon_0} \log _e \frac{1}{2}\)

Solution: 1. \(\frac{\mathrm{q} \lambda}{\pi \varepsilon_0} \log _{\mathrm{e}} \sqrt{2}\)

Question 22. A flat circular fixed disc has a charge +Q uniformly distributed on the disc. A charge +q is thrown with kinetic energy K, towards the disc along its axis. The charge q :

1. May hit the disc at the centre
2. May return along its path after touching the disc
3. May return along its path without touching the disc
4. Any of the above three situations is possible depending on the magnitude of K

Solution: 4. Any of the above three situations is possible depending on the magnitude of K

Section (5): Potential Energy Of A System Of Point Charge

Question 1. In the H atom, an electron is rotating around the proton in an orbit of radius r. Work done by an electron in moving once around the proton along the orbit will be –

1. ke/r
2. ke2/r²
3. 2πre
4. zero

Solution: 4. zero

Question 2. You are given an arrangement of three point charges q, 2q, and xq separated by equal finite distances so that the electric potential energy of the system is zero. Then the value of x is :

1. \(-\frac{2}{3}\)
2. \(-\frac{1}{3}\)
3. \(\frac{2}{3}\)
4. \(\frac{3}{2}\)

Solution: 1. \(-\frac{2}{3}\)

Question 3. You are given an arrangement of three point charges q, 2q, and xq separated by equal finite distances so that the electric potential energy of the system is zero. Then the value of x is :

1. \(-\frac{2}{3}\)
2. \(-\frac{1}{3}\)
3. \(\frac{2}{3}\)
4. \(\frac{3}{2}\)

Solution: 1. \(-\frac{2}{3}\)

Question 4. If a charge q is placed at the center of the line joining two equal charges Q each such that the system is in equilibrium, then the value of q is :

1. Q / 2
2. –Q/2
3. Q / 4
4. –Q/4

Solution: 4. –Q/4

Section (6): Self Energy And Energy Density

Question 1. A sphere of radius 1 cm has a potential of 8000 V. The energy density near the surface of the sphere will be:

1. 64 × 10⁵ J/m³
2. 8 × 10³ J/m³
3. 32 J/m³
4. 2.83 J/m³

Solution: 4. 2.83 J/m³

Question 2. If ‘ n ‘ identical water drops assumed spherical each charged to a potential energy U coalesce to a single drop, the potential energy of the single drop is(Assume that drops are uniformly charged):

1. n₁/3 U
2. n₂/3 U
3. n₄/3 U
4. n₅/3 U

Solution: 4. n₅/3 U

Question 3. Four charges equal to –Q each are placed at the four corners of a square and a charge q is at its center. If the system is in equilibrium, the value of q is:

1. \(-\frac{Q}{4}(1+2 \sqrt{2})\)
2. \(\frac{Q}{4}(1+2 \sqrt{2})\)
3. \(-\frac{Q}{2}(1+2 \sqrt{2})\)
4. \(\frac{Q}{2}(1+2 \sqrt{2})\)

Solution: 2. \(\frac{Q}{4}(1+2 \sqrt{2})\)

Section (7): Questions Based On Relation Between \(\overrightarrow{\mathrm{E}}\) And V :

Question 1. A family of equipotential surfaces is shown. The direction of the electric field at point A is along- –

1. AB
2. AC
3. AD
4. AF

Solution: 4. AF

Question 2. Some equipotential surfaces are shown in the figure. The magnitude and direction of the electric field is-

1. 100 V/m making angle 1200 with the x-axis
2. 100 V/m making angle 600 with the x-axis
3. 200 V/m making angle 1200 with the x-axis
4. None of the above

Solution: 3. 200 V/m making angle 1200 with the x-axis

Question 3. The variation of potential with distance r from a fixed point is shown in Figure. The electric field at r = 5 cm, is :

1. (2.5) V/cm
2. (–2.5) V/cm
3. (–2/5) cm
4. (2/5) V/cm

Solution: 1. (2.5) V/cm

Question 4. The electric field and the electric potential at a point are E and V respectively

1. If E = 0, V must be zero
2. If V = 0, E must be zero
3. If E ≠ 0, V cannot be zero
4. None of these

Solution: 4. None of these

Question 5. The electric field in a region is directed outward and is proportional to the distance r from the origin. Taking the electric potential at the origin to be zero, the electric potential at a distance r :

1. Is uniform in the region
2. Is proportional to r
3. Is proportional to r²
4. Increases as one goes away from the origin.

Solution: 3. Is proportional to r²

Question 6. V = any, then the electric field at a point will be proportional to :

1. r
2. r^-1
3. r^-2
4. r²

Solution: 1. r

Question 7. A point charge is located at O. There is a point P at a distance r from it. The electric field at point P is 500 V/m and has a potential of 3000 V. Then the value of r is

1. 6 m
2. 12 m
3. 24 m
4. 36 m

Solution: 1. 6 m

Question 8. Figure shows three points A, B, and C in a region of uniform electric field \(\overrightarrow{\mathrm{E}}\). The line AB is perpendicular and BC is parallel to the field lines. Then which of the following holds good? 

1. V_A= V_B= V_C
2. V_A= V_B> V_C
3. V_A= V_B< V_C
4. V_A> V_B= V_C

where V_A> V_Band represents the electric potential at points A, B, and C respectively.

Solution: 2. V_A= V_B> V_C

Question 9. The variation of potential with distance r from a fixed point is shown in the figure. The electric field at r = 3 cm is :

1. Zero
2. –2.5 V/cm
3. +2.5 V/cm
4. +5 V/cm

Solution: 1. Zero

Question 10. Electric potential at any point is V = –5x + 3y + √15z, then the magnitude of the electric field is –

1. 3 √2
2. 4 √2
3. 5√2
4. 7

Solution: 4. 7

Question 11. The potential at a point x (measured in µm) due to some charges situated on the x-axis is given by V(x) = 20/(x² – 4) volts. The electric field E at x = 4 μm is given by :

1. 5/3 volt/µm and in the –ve x direction
2. 5/3 volt/µm and in the +ve x direction
3. 10/9 volt/µm and in the –ve x direction
4. 10/9 volt/µm and in the +ve x direction

Solution: 4. 10/9 volt/µm and in the +ve x direction

Question 12. The electric potential V as a function of distance x (in meters) is given by V = (5x² + 10x ^-9) volt.

The value of the electric field at x = 1 m would be :

1. – 20 volt/m
2. 6 volt/m
3. 11 volt/m
4. –23 volt/m

Solution: 1. – 20 volt/m

Question 13. A uniform electric field having a magnitude and direction along a positive X-axis exists. If the electric potential V is zero at x = 0, then its value at x = +x will be :

1. V_X= xE₀
2. V_X= –xE₀
3. V_X= x²E₀
4. V_X= –x² E₀

Solution: 2. V_X= –xE₀

Question 14. A uniform electric field pointing in a positive x-direction exists in a region. Let A be the origin, B be the point on the x-axis at x = + 1 cm and C be the point on the y-axis at y = + 1 cm. Then the potentials at points A, B, and C satisfy :

V_A< V_B

V_A> V_B

V_A< V_C

V_A> V_C

Solution: 2. V_A> V_B

Question 15. A 5-coulomb charge experiences a constant force of 2000 N when moved between two points separated by a distance of 2 cm in a uniform electric field. The potential difference between these two points is:

1. 8 V
2. 200 V
3. 800 V
4. 20,000 V

Solution: 1. 8 V

Question 16. An equipotential surface and an electric line of force :

1. Never intersect each other
2. Intersect at 45º
3. Intersect at 60º
4. Intersect at 90º

Solution: 4. Intersect at 90º

Section (8): Dipole

Question 1. If an electric dipole is kept in a non-uniform electric field, then it will experience –

1. Only torque
2. No torque
3. A resultant force and a torque
4. Only a force

Solution: 3. A resultant force and a torque

Question 2. The force on a charge situated on the axis of a dipole is F. If the charge is shifted to double the distance, the acting force will be –

1. 4F
2. F/2
3. F/4
4. F/8

Solution: 4. F/8

Question 3. A dipole of dipole moment p is placed in an electric field \(\vec{E}\) and is in stable equilibrium. The torque required to rotate the dipole from this position by angle θ will be

1. pE cos θ
2. pE sin θ
3. pE tan θ
4. –pE cosθ

Solution: 2. pE sin θ

Question 4. The electric potential at a point due to an electric dipole will be –

1. \(\frac{k(\vec{p} \cdot \vec{r})}{r^3}\)
2. \(\frac{k(\vec{p} \cdot \vec{r})}{r^2}/\)
3. \(\frac{k(\vec{p} \times \vec{r})}{r}\)
4. \(\frac{k(\overrightarrow{\mathrm{p}} \times \overrightarrow{\mathrm{r}})}{r^2}\)

Solution: 1. \(\frac{k(\vec{p} \cdot \vec{r})}{r^3}\)

Question 5. The ratio of electric fields due to an electric dipole on the axis and the equatorial line at an equal distance will be –

1. 4: 1
2. 1: 2
3. 2: 1
4. 1: 1

Solution: 3. 2: 1

Question 6. An electric dipole is made up of two equal and opposite charges of 2 x 10^-6 coulomb at a distance of 3 cm. This is kept in an electric field of 2 x 105 N/C, then the maximum torque acting on the dipole –

1. 12 x 10^-1 Nm
2. 12 x 10^-3 Nm
3. 24 x 10^-3 Nm
4. 24 x 10^-1 Nm

Solution: 2. 12 x 10^-3 Nm

Question 7. The distance between two singly ionized atoms is 1Å. If the charge on both ions is equal and opposite then the dipole moment in coulomb-metre is –

1. 1.6 × 10^-29
2. 0.16 × 10^-29
3. 16 × 10^-29
4. 1.6 × 10^-29/ 4πε₀

Solution: 1. 1.6 × 10^-29

Question 8. The electric potential in volts at a distance of 0.01 m on the equatorial line of an electric dipole of dipole moment p is –

1. \(\mathrm{p} / 4 \pi \epsilon_0 \times 10^{-4}\)
2. zero
3. \(4 \pi \epsilon_0 \mathrm{p} \times 10^{-4}\)
4. \(4 \pi \epsilon_0 / p \times 10^{-4}\)

Solution: 2. zero

Question 9. The electric potential in volts due to an electric dipole of dipole moment 2 x 10^-8 C-m at a distance of 3m on a line making an angle of 600 with the axis of the dipole is –

1. 0
2. 10
3. 20
4. 40

Solution: 2. 10

Question 10. A dipole of electric dipole moment P is placed in a uniform electric field of strength E. If θ is the angle between positive directions of P and E, then the potential energy of the electric dipole is largest when θ is :

1. zero
2. π/2
3. π
4. π/4

Solution: 3. π

Question 11. Potential due to an electric dipole at some point is maximum or minimum when the axis of the dipole and line joining point and dipole are at angles respectively :

1. 90° and 180°
2. 0° and 90°
3. 90° and 0°
4. 0° and 180°

Solution: 4. 0° and 180°

Question 12. The electric field on the axis of an electric dipole, at a distance of r from its center, is E. If the dipole is rotated through 90°; then the electric field intensity at the same point will be :

1. E
2. \(\frac{E}{4}\)
3. \(\frac{E}{2}\)
4. 2E

Solution: 3. \(\frac{E}{2}\)

Question 13. An electric dipole is placed along a North-South direction in a sphere filled with water. Which statement is true?

1. Electric flux is coming towards the sphere.
2. Electric flux is going out of the sphere
3. As much electric flux is going out of the sphere, as much is coming toward the sphere.
4. Water does not allow electric flux to come inside the sphere

Solution: 3. As much electric flux is going out of the sphere, as much is coming toward the sphere.

Question 14. At the equator of an electric dipole, the angle between the electric dipole moment and an electric field is :

1. 0°
2. 90°
3. 180°
4. None of these

Solution: 3. 180°

Question 15. The potential of the dipole at its axial position is proportional to distance r as :

1. r^-2
2. r^-1
3. r
4. r0

Solution: 1. r^-2

Question 16. An electric dipole has the magnitude of its charge as q and its dipole moment is p. It is placed in a uniform electric field E. If its dipole moment is along the direction of the field, the force on it and its potential energy are respectively :

1. 2qE and minimum
2. qe and pE
3. zero and minimum
4. qE and maximum

Solution: 3. zero and minimum

Question 17. An electric dipole of moment dipole by 90° is pis lying along a uniform electric field \(\vec{E}\). The work done in rotating the

1. √2pE
2. \(\frac{\mathrm{pE}}{2}\)
3. 2pE
4. pE

Solution: 4. pE

Question 18. Three point charges +q, –2q and + q are placed at points (x = 0, y = a, z = 0), (x = 0, y = 0, z = 0) and (x = a, y = 0, z = 0), respectively. The magnitude and direction of the electric dipole moment vector of this charge assembly are :

1. v2qa along + y direction
2. 2qa along the line joining points (x = 0, y = 0, z = 0) and (x = a, y = a, z = 0)
3. qa along the line joining points (x = 0, y = 0, z = 0) and (x = a, y = a, z = 0)
4. 2qa along + x direction

Solution: 2. 2qa along the line joining points (x = 0, y = 0, z = 0) and (x = a, y = a, z = 0)

Question 19. An electric dipole is placed at an angle of 30 to a non-uniform electric field. The dipole will experience

1. A torque as well as a translational force.
2. A torque only.
3. A translational force only in the direction of the field.
4. A translational force is only in a direction normal to the direction of the field.

Solution: 1. A torque as well as a translational force.

Question 20. Due to an electric dipole shown in fig., the electric field intensity is parallel to the dipole axis :

1. At P only
2. A Q only
3. Both at P and at Q
4. Neither at P nor at Q

Solution: 3. Both at P and at Q

Question 21. An electric dipole consists of two opposite charges each of magnitude 1.0 μC, separated by a distance of 2.0 cm. The dipole is placed in an external electric field of 1.0 × 10⁵ N/C. The maximum torque on the dipole is :

1. 0.2 × 10^-3 N-m
2. 1.0 × 10^-3 N-m
3. 2.0 × 10^-3 N-m
4. 4.0 × 10^-3 N-m

Solution: 3. 2.0 × 10^-3 N-m

Question 22. Two opposite and equal charges of magnitude 4 × 10^-8 coulomb each when placed 2 × 10^-5 cm apart form a dipole. If this dipole is placed in an external electric field of 4 × 10⁸ N/C, the value of maximum torque and the work required in rotating it through 180º from its initial orientation which is along the electric field will be : (Assume rotation of dipole about an axis passing through the center of the dipole):

1. 64 × 10^-4 N-m and 44 × 10^-4 J
2. 32 × 10^-4 N-m and 32 × 10^-4J
3. 64 × 10^-4 N-m and 32 × 10^-4 J
4. 32 × 10^-4 N-m and 64 × 10^-4 J

Solution: 4. 32 × 10^-4 N-m and 64 × 10^-4J

Question 23. At a point on the axis (but not inside the dipole and not at infinity) of an electric dipole

1. The electric field is zero
2. The electric potential is zero
3. Neither the electric field nor the electric potential is zero
4. The electric field is directed perpendicular to the axis of the dipole

Solution: 3. Neither the electric field nor the electric potential is zero

Section (9): Flux Calculation And Gauss’s Law

Question 1. For an electrostatic system which of the statements is always true :

1. Electric lines are parallel to metallic surfaces.
2. The electric field inside a metallic surface is zero.
3. Electric lines of force are perpendicular to the equipotential surface.
   1. (1) and (2) only
   2. (2) and (3) only
   3. (1) and (3) only
   4. (1), (2), and (3)

Solution: 3. (1) and (3) only

Question 2. Total flux coming out of some closed surface is :

1. q/ε₀
2. ε₀/q
3. qε₀
4. Vq/ ε0

Solution: 4. Vq/ ε₀

Question 3. Three charges q1= 1 × 10^-6, q2= 2 × 10^-6, q3= –3 × 10^-6 C have been placed, as shown in the figure, in four surfaces S₁, S₂, S₃and S₄electrical flux emitted from the surface S₂ in N–m2/C will be –

1. 36π × 10³
2. –36π × 10³
3. 36π × 10⁹
4. –36π × 10⁹

Solution: 2. –36π × 10³

Question 4. The intensity of an electric field at some point distant r from the axis of an infinitely long pipe having charges per unit length as q will be :

1. Proportional to r2
2. Proportional to r3
3. Inversely proportional to r.
4. Inversely proportional to r2.

Solution: 1. Proportional to r2

Question 5. Eight charges, 1μC, -7μC, -4μC, 10μC, 2μC, -5μC, -3μC and 6μC are situated at the eight corners of a cube of side 20 cm. A spherical surface of radius 80 cm encloses this cube. The center of the sphere coincides with the center of the cube. Then the total outgoing flux from the spherical surface (in units of volt meter) is-

1. 36π x 10³
2. 684π x 10³
3. zero
4. None of the above

Solution: 3. zero

Question 6. A closed cylinder of radius R and length L is placed in a uniform electric field E, parallel to the axis of the cylinder. Then the electric flux through the cylinder must be –

1. 2πR²E
2. (2πR² + 2πRL)E
3. 2πRLE
4. zero

Solution: 4. zero

Question 7. A charge q is placed at the center of the cubical vessel (with one face open) as shown in the figure. The flux of the electric field through the surface of the vessel is

1. zero
2. q/ε₀
3. \(\frac{\mathrm{q}}{4 \varepsilon_0}\)
4. 5q/6ε₀

Solution: 4. 5q/6ε₀

Question 8. Electric charge is uniformly distributed along a long straight wire of radius 1 mm. The charge per cm length of the wire is Q coulomb. Another cylindrical surface of radius 50 cm and length 1m symmetrically encloses the wire as shown in Fig. The total electric flux passing through the cylindrical surface is –

1. \(\frac{\mathrm{Q}}{\varepsilon_0}/\)
2. \(\frac{100 Q}{\varepsilon_0}\)
3. \(\frac{10 Q}{\left(\pi \varepsilon_0\right)}\)
4. \(\frac{100 Q}{\left(\pi \varepsilon_0\right)}\)

Solution: 2. \(\frac{100 Q}{\varepsilon_0}\)

Question 9. If the geometric axis of the cylinder is parallel to the electric field, the flux through the cylinder will be –

1. 2πr × B
2. πr² × B
3. 2πr² × B
4. 0

Solution: 4. 0

Question 10. A cubical box contains charge +Q at its center. The total electric flux emerging from the box is :

1. \(\frac{\mathrm{Q}}{\varepsilon_0}\)
2. \(\frac{\mathrm{Q}}{6 \varepsilon_0}\)
3. \(\frac{\mathrm{Q}}{4 \varepsilon_0}\)
4. \(\varepsilon_0 \mathrm{Q}\)

Solution: 1. \(\frac{\mathrm{Q}}{\varepsilon_0}\)

Question 11. If a square coil is making an angle of 60° with electric field E according to the figure, the electric flux passing through the square coil is (the side of the square is 4 cm) :

1. 853 E
2. 8E
3. 16 E
4. None of these

Solution: 2. 8E

Question 12. Four equal charges q are placed at the center of a conducting hollow sphere. If they are displaced 1.5 cm from the center, the change in flux will be (Radius of sphere> 1.5 cm) :

1. Doubled
2. Rripled
3. Constant
4. None of these

Solution: 4. None of these

Question 13. If the electric flux entering and leaving an enclosed surface respectively is φ1and φ2, the electric charge inside the surface will be

1. (φ2– φ1) ε₀
2. (φ1+ φ2)/ε₀
3. (φ2– φ1)/ ε₀
4. (φ1+ φ2) ε₀

Solution: 1. (φ2– φ1) ε₀

Question 14. A square surface of side L m is in the plane of the paper. A uniform electric field \(\overrightarrow{\mathrm{E}}\)(V/m), also in the plane of the paper, is limited only to the lower half of the square surface, (see figure). The electric flux in SI units associated with the surface is :

1. EL2/ (2ε₀)
2. EL2/ 2
3. zero
4. EL2

Solution: 3. zero

Question 15. If the electric flux entering and leaving an enclosed surface respectively is φ1and φ2, the electric charge inside the surface will be :

1. (φ2– φ1)ε₀
2. (φ1+ φ2)/ε₀
3. (φ2– φ1)/ε₀
4. (φ1+ φ2) ε₀

Solution: 1. (φ2– φ1)ε₀

Question 16. An electric dipole is placed at the center of a sphere. Mark the correct options.

1. The electric field is zero at every point of the sphere.
2. The flux of the electric field through the sphere is non-zero.
3. The electric field is zero on a circle on the sphere.
4. The electric field is not zero anywhere on the sphere.

Solution: 4. The electric field is not zero anywhere on the sphere.

Question 17. A charge Q is placed at a distance of 4R above the center of a disc of radius R. The magnitude of flux through the disc is φ. Now a hemispherical shell of radius R is placed over the disc such that it forms a closed surface. The flux through the curved surface (taking the direction of the area vector along outward normal as positive), is –

1. Zero
2. φ
3. – φ
4. 2φ

Solution: 3. – φ

Question 18. A charge q is placed at the corner of a cube of side a. The electric flux through the cube is :

1. \(\frac{q}{\varepsilon_0}\)
2. \(\frac{q}{3 \varepsilon_0}\)
3. \(\frac{q}{6 \varepsilon_0}\)
4. \(\frac{q}{8 \varepsilon_0}\)

Solution: 4. \(\frac{q}{8 \varepsilon_0}\)

Question 19. A charge qμC is placed at the center of a cube of a side 0.1 m, then the electric flux diverging from each face of the cube is :

1. \(\frac{\mathrm{q} \times 10^{-6}}{24 \varepsilon_0}\)
2. \(\frac{\mathrm{q} \times 10^{-4}}{\varepsilon_0}\)
3. \(\frac{\mathrm{q} \times 10^{-6}}{6 \varepsilon_0}\)
4. \(\frac{\mathrm{q} \times 10^{-4}}{12 \varepsilon_0}\)

Solution: 3. \(\frac{\mathrm{q} \times 10^{-6}}{6 \varepsilon_0}\)

Question 20. Gauss law is given by ∈0 \(\int \vec{E} \cdot \overrightarrow{d s}=q\) if net charge enclosed by Gaussian surface is zero then –

1. E on the surface must be zero
2. Incoming and outgoing electric lines are equal
3. There is a net incoming electric flux
4. None

Solution: 2. Incoming and outgoing electric lines are equal

Section (10): Conductor, Its Properties & Electric Pressure

Question 1. The electric field near the conducting surface of a uniform charge density σ will be –

1. σ / ∈0and parallel to the surface.
2. 2σ /∈0and parallel to surface.
3. σ / ∈0and perpendicular to the surface.
4. 2σ / ∈0 and perpendicular to the surface.

Solution: 3. σ / ∈0and perpendicular to the surface.

Question 2. An uncharged conductor A is brought close to another positive charged conductor B, then the charge on B –

1. Will increase but potential will be constant.
2. Will be constant but the potential will increase
3. Will be constant but the potential decreases.
4. The potential and charge on both are constant.

Solution: 3. Will be constant but potential decreases.

Question 3. The fig. shows lines of constant potential in a region in which an electric field is present. The value of the potential is written in brackets of the points A, B, and C, the magnitude of the electric field is greatest at point –

1. A
2. B
3. C
4. A and C

Solution: 2. B

Question 4. The electric charge in uniform motion produces –

1. An electric field only
2. A magnetic field only
3. Both electric and magnetic fields
4. Neither electric nor magnetic fields

Solution: 3. Both electric and magnetic fields

Question 5. Which of the following represents the correct graph for electric field intensity and the distance r from the center of a hollow charged metal sphere or solid metallic conductor of radius R :

Solution: 4.

Question 6. A neutral metallic object is placed near a finite metal plate carrying a positive charge. The electric force on the object will be :

1. Towards the plate
2. A way from the plate
3. Parallel to the plate
4. Zero

Solution: 1. Towards the plate

Question 7. The figure shows a thick metallic sphere. If it is given a charge +Q, then an electric field will be present in the region

1. r < R₁only
2. r > R₁and R₁< r < R₂
3. r ≥R₂only
4. r≤R₂only

Solution: 3. r ≥R2only

Question 8. An uncharged sphere of metal is placed in a uniform electric field produced by two large conducting parallel plates having equal and opposite charges, then lines of force look like

Solution: 3.

Question 9. You are traveling in a car during a thunderstorm, to protect yourself from lightning would you prefer to :

1. Remain in the car
2. Take shelter under a tree
3. Get out and be flat on the ground
4. Touch the nearest electrical pole

Solution: 1. Remain in the car

Question 10. The amount of work done in Joules in carrying a charge +q along the closed path PQRSP between the oppositely charged metal plates is (where E is the electric field between the plates)

1. zero
2. q
3. qE (PQ + QR + SR + SP)
4. q\ε₀

Solution: 1. zero

Question 11. The figure shows a closed surface that intersects a conducting sphere. If a positive charge is placed at the point P, the flux of the electric field through the closed surface

1. Will remain zero
2. Will become positive
3. Will become negative
4. Will become undefined

Solution: 2. Will become positive

Question 12. A charge ‘ q ‘ is placed at the center of a conducting spherical shell of radius R, which is given a charge Q. An external charge Q′ is also present at distance R′ (R′ > R) from ‘ q ‘. Then the resultant field will be best represented for region r < R by: [ where r is the distance of the point from q ]

Solution: 1.

Question 13. In the above question, if Q’ is removed then which option is correct :

Solution: 1.

Question 14. The net charge is given to an isolated conducting solid sphere:

1. must be distributed uniformly on the surface
2. may be distributed uniformly on the surface
3. must be distributed uniformly in the volume
4. may be distributed uniformly in the volume.

Solution: 1. must be distributed uniformly on the surface

Question 15. The net charge is given to a solid insulating sphere:

1. must be distributed uniformly in its volume
2. may be distributed uniformly in its volume
3. must be distributed uniformly on its surface
4. the distribution will depend upon whether other charges are present or not.

Solution: 2. may be distributed uniformly in its volume

Question 16. A charge Q is kept at the center of a conducting sphere of inner radius and outer radius R2. A point charge q is kept at a distance r (> R2) from the center. If q experiences an electrostatic force of 10 N then assuming that no other charges are present, the electrostatic force experienced by Q will be:

1. –10 N
2. 0
3. 20 N
4. None of these

Solution: 2. 0

Question 17. A solid conducting sphere having a charge Q is surrounded by an uncharged concentric conducting hollow spherical shell. Let the potential difference between the surface of the solid sphere and that of the outer surface of the hollow shell be V. If the shell is now given a charge –3Q, the new potential difference between the same two surfaces is :

1. V
2. 2V
3. 4V
4. –2V

Solution: 1. V

Question 18. A point charge ‘ q ‘ is placed at a point inside a hollow conducting sphere. Which of the following electric force patterns is correct?

Solution: 1.

Question 19. Consider a neutral conducting sphere. A positive point charge is placed outside the sphere. The net charge on the sphere is then,

1. Negative and distributed uniformly over the surface of the sphere
2. Negative and appears only at the point on the sphere closest to the point charge
3. Negative and distributed non-uniformly over the entire surface of the sphere
4. Zero

Solution: 4. Zero

Question 20. Three concentric metallic spherical shells of radii R, 2R, and 3R, are given charges Q₁, Q₂, and Q₃, respectively. It is found that the surface charge densities on the outer surfaces of the shells are equal. Then, the ratio of the charges given to the shells, Q₁: Q₂: Q₃, is

1. 1: 2 : 3
2. 1 : 3: 5
3. 1: 4: 9
4. 1: 8: 18

Solution: 2. 1 : 3: 5

Question 21. A positive point charge q is brought near a neutral metal sphere.

1. The sphere becomes negatively charged.
2. The sphere becomes positively charged.
3. The interior remains neutral and the surface gets non-uniform charge distribution.
4. The interior becomes positively charged and the surface becomes negatively charged.

Solution: 3. The interior remains neutral and the surface gets non-uniform charge distribution.

Question 22. Two small conductors A and B are given charges q₁ and respectively. Now they are placed inside a hollow metallic conductor (C) carrying a charge Q. If all the three conductors A, B, and C are connected by conducting wires as shown, the charges on A, B, and C will be respectively:

1. \( \frac{q_1+q_2}{2}, \frac{q_1+q_2}{2}, Q\)
2. \(\frac{Q+q_1+q_3}{3}, \frac{Q+q_1+q_2}{3}, \frac{Q+q_1+q_2}{3}\)
3. \(\frac{q_1+q_2+Q}{2}, \frac{q_1+q_2+Q}{2}, 0\)
4. 0, 0, Q + q₁ + q₂

Solution: 4. 0, 0, Q + q₁+ q₂

Question 23. A charge Q is kept at the center of a conducting sphere of inner radius R₁ and outer radius R₂. A point charge q is kept at a distance r (> R2) from the center. If q experiences an electrostatic force of 10 N then assuming that no other charges are present, the electrostatic force experienced by Q will be:

1. –10 N
2. 0
3. 20 N
4. None of these

Solution: 2. 0

Question 24. Some charge is being given to a conductor then its potential is :

1. Maximum at surface
2. Maximum at centre
3. The same throughout the conductor
4. Maximum somewhere between surface and center

Solution: 3. Same throughout the conductor

Question 25. A solid metallic sphere has a charge of +3Q. Concentric with this sphere is a conducting spherical shell having charge –Q. The radius of the sphere is a and that of the spherical shell is b(>a). What is the electric field at a distance r(a < r < b) from the center?

1. \(\frac{1}{4 \pi \varepsilon_0} \frac{\mathrm{Q}}{\mathrm{r}}\)
2. \(\frac{1}{4 \pi \varepsilon_0} \frac{3 Q}{r}\)
3. \(\frac{1}{4 \pi \varepsilon_0} \frac{3 \mathrm{Q}}{\mathrm{r}^2}\)
4. \(\frac{1}{4 \pi \varepsilon_0} \frac{Q}{r^2}\)

Solution: 3. \(\frac{1}{4 \pi \varepsilon_0} \frac{3 \mathrm{Q}}{\mathrm{r}^2}\)

Question 26. Two charged spheres having radii a and b are joined with a wire then the ratio of electric field Ea/Ebon on their surface is –

1. a/b
2. b/a
3. a²/b²
4. b²/a²

Solution: 2. b/a

Question 27. A long hollow conducting cylinder is kept coaxially inside another long, hollow conducting cylinder of a larger radius. Both the cylinders are initially electrically neutral.

1. A potential difference appears between the two cylinders when a charge density is given to the inner cylinder.
2. A potential difference appears between the two cylinders when a charge density is given to the outer cylinder.
3. No potential difference appears between the two cylinders when a uniform line charge is kept along the axis of the cylinders.
4. No potential difference appears between the two cylinders when the same charge density is given to both cylinders.

Solution: 1. A potential difference appears between the two cylinders when a charge density is given to the inner cylinder.

Electrostatics Exercise – 2

Question 1. A dipole having dipole moment p is placed in front of a solid uncharged conducting sphere as shown in the diagram. The net potential at point A lying on the surface of the sphere is :

1. \(\frac{k p \cos \phi}{r^2}\)
2. \(\frac{\mathrm{kpcos}^2 \phi}{\mathrm{r}^2}\)
3. Zero
4. \(\frac{2 \mathrm{kp} \cos ^2 \phi}{\mathrm{r}^2}\)

Solution: 2. \(\frac{\mathrm{kpcos}^2 \phi}{\mathrm{r}^2}\)

Question 2. Two equal charges are separated by a distance d. A third charge placed on a perpendicular bisector at x distance will experience maximum coulomb force when –

1. x = d / √2
2. x = d/2
3. x = d/2√2
4. x = d/2 v3

Solution: 3. x = d/2 √2

Question 3. The work done in placing four charges at the corners of a square as shown in the figure, will be –

1. \( (4-\sqrt{2}) \frac{K q^2}{a}\)
2. \((4+\sqrt{2}) \frac{K q^2}{\mathrm{a}}\)
3. \((4-\sqrt{2}) \frac{K q^2}{a^2}\)
4. \((4+\sqrt{2}) \frac{K q^2}{\mathrm{a}^2}\)

Solution: 1. \( (4-\sqrt{2}) \frac{K q^2}{a}\)

Question 4. Six charges q,q,q, – q, –q, and –q to be arranged on the vertices of a regular hexagon PQRSTU such that the electric field at the center is double the field produced when only charge ‘q’ is placed at vertex R. The sequence of the charges from P to U is

1. q, –q, q, q, –q, –q
2. q, q, q, –q, –q, –q
3. –q, q, q, –q, –q, q
4. –q, q, q, q, –q, –q

Solution: 1. q, –q, q, q, –q, –q

Question 5. Which of the following groups do not have the same dimensions

1. Young’s modulus, pressure, stress
2. work, heat, energy
3. Electromotive force, potential difference, voltage
4. Electric dipole, electric flux, electric field

Solution: 4. Electric dipole, electric flux, electric field

Question 6. A spherical portion has been removed from a solid sphere having a charge distributed uniformly in its volume as shown in the figure. The electric field inside the emptied space is

1. Zero everywhere
2. Is not zero but uniform
3. Nonuniform
4. Is zero at the center only

Solution: 2. Is not zero but uniform

Question 7. Statement -1: For practical purposes, the earth is used as a reference at zero potential in electrical circuits.

Statement -2: The electrical potential of a sphere of radius R with charge Q uniformly distributed on the surface is given by \(\)

1. Statement -1 is True, Statement -2 is True; Statement -2 is a correct explanation for Statement -1
2. Statement -1 is True, Statement -2 is True; Statement -2 is NOT a correct explanation for Statement -1
3. Statement -1 is True, Statement -2 is False
4. Statement -1 is False, Statement -2 is True.

Solution: 2. Statement -1 is True, Statement -2 is True; Statement -2 is NOT a correct explanation for Statement -1

Question 8. Which of the following statement(s) is/are correct?

1. If the electric field due to a point charge varies as r –2.5 instead of r –2, then the Gauss law will still be valid.
2. The Gauss law can be used to calculate the field distribution around an electric dipole.
3. If the electric field between two point charges is zero somewhere, then the sign of the two charges is the same.
4. The work done by the external force in moving a unit positive charge from point A at potential VA to point B at potential VBis (VB — VA).

Solution: 3. If the electric field between two point charges is zero somewhere, then the sign of the two charges is the same.

Question 9. Let E1= x ˆ i+ y ˆ j ,and E2= xy2 ˆ i+ x2y ˆ j, then :

1. Represents a constant electric field
2. Represents a constant electric field
3. Both represent a constant electric field
4. None of these

Solution: 4. None of these

Question 10. When a glass rod is rubbed with silk, the amount of positive charge acquired by the glass rod in magnitude is :

1. Less than the charge for silk
2. Greater than the charge on silk
3. Equal to the charge on silk
4. None of these

Solution: 3. Equal to the charge on silk

Question 11. A cube has point charges of magnitude – q at all its vertices. The electric field at the center of the cube is :

1. \(1) \frac{1}{4 \pi \varepsilon_0} \frac{6 q}{3 a^2}\)
2. \(\frac{1}{4 \pi \varepsilon_0} \frac{8 \mathrm{q}}{\mathrm{a}^2}\)
3.  zero
4. \(\frac{1}{4 \pi \varepsilon_0} \frac{-8 \mathrm{q}}{\mathrm{a}^2}\)

Solution: 3.

Question 12. Three point charges +q, – 2q and +q placed at points (x = 0, y = a, z = 0), (x = 0, y = 0, z = 0) and (x = a, y = 0, z = 0), respectively. The magnitude and direction of the electric dipole moment vector of this charge assembly are

1. √2qa along +y direction
2. √2qa along the line joining points (x = 0, y = 0, z = 0) and (x = a, y = a, z = 0) (2)
3. qa along the line joining points (x = 0, y = 0, z = 0) and (x = a, y = a, z = 0)
4. √2qa along +x direction

Solution: 2. v2qa along the line joining points (x = 0, y = 0, z = 0) and (x = a, y = a, z = 0) (2)

Question 13. An electric dipole is placed along the x-axis at the origin O. A point P is at a distance of 20 cm from this origin such that OP makes an angle π/3 with the x-axis. If the electric field at P makes an angle θ with the x-axis, the value of θ would be

1. \(\frac{\pi}{3}\)
2. \(\frac{\pi}{3}+\tan ^{-1}\left(\frac{\sqrt{3}}{2}\right)\)
3. \(\frac{2 \pi}{3}\)
4. \(\tan ^{-1}\left(\frac{\sqrt{3}}{2}\right)\)

Solution: 2. \(\frac{\pi}{3}+\tan ^{-1}\left(\frac{\sqrt{3}}{2}\right)\)

Question 14. A Charged wire is bent in the form of a semi-circular arc of radius a. If charge per unit length is λ coulomb/meter, the electric field at the center O is :

1. Zero
2. \(\frac{\lambda}{2 \pi \mathrm{a}^2 \varepsilon_0}\)
3. \(\frac{\lambda}{4 \pi^2 \varepsilon_0 a}\)
4. \(\frac{\lambda}{2 \pi \varepsilon_0 \mathrm{a}}\)

Solution: 3. \(\frac{\lambda}{4 \pi^2 \varepsilon_0 a}\)

Question 15. The dimensions of \(\)(ε₀: permittivity of free space; E: electric field) are: 2

1. M L T^-1
2. M L² T^-2
3. M L T^-2
4. M L^-1 T^-2

Solution: 4. M L^-1T^-2

Question 16. Two non–non-conducting spheres of radii R₁ and drying uniform volume charge densities +ρ and – ρ, respectively, are placed such that they partially overlap, as shown in the figure. At all points in the overlapping region :

1. The electrostatic field is zero
2. The electrostatic potential is constant
3. The electrostatic field is constant
4. The electrostatic field has the same magnitude only

Solution: 3. The electrostatic field is constant

Question 17. Charges Q₁, 2Q, and 4Q are uniformly distributed in three dielectric solid spheres 1, 2, and 3 of radii R/2, R, and 2R respectively, as shown in the figure. If magnitudes of the electric fields at point P at a distance R from the center of spheres 1, 2, and 3 are E1 and respectively, then

1. E₁> E₂> E₃
2. E₃> E₁> E₂
3. E₂> E₁> E₃
4. E₃> E₂> E₁

Solution: 3. E₂> E1> E₃

Question 18. Four charges Q₁, Q₂, Q₃, and Q₄, of the same magnitude, are fixed along the x-axis at x = –2a –a, +a, and +2a, respectively. A positive charge q is placed on the positive y-axis at a distance b > 0. Four options for the signs of these charges are given in List I. The direction of the forces on the charge q is given in List- II Match List-1 with List-II and select the correct answer using the code given below the lists.

List-I   List-II

P. Q₁,Q2,Q₃, Q₄, all positive 1. +x

Q. Q₁,Q₂positive Q₃,Q₄ negative 2. –x

R. Q₁,Q₄positive Q₂, Q₃negative 3. +y

S. Q₁,Q₃positive Q₂, Q₄negative 4. –y

Code :

1. P-3, Q-1, R-4,S-2
2. P-4, Q-2, R-3, S-1
3. P-3, Q-1, R-2,S-4
4. P-4, Q-2, R-1, S-3

Solution: 1. P-3, Q-1, R-4,S-2

Electrostatics Exercise – 3

Question 1. The electric potential at a point (x, y, z) is given by V = – x2 y – xz3 + 4 The electric field \(\text { है }\) at that point is

1. \(\vec{E}=\hat{i}\left(2 x y+z^3\right)+\hat{j} x^2+\hat{k} 3 x z^2\)
2. \(\vec{E}=\hat{i} 2 x y+\hat{j}\left(x^2+y^2\right)+\hat{k}\left(3 x z-y^2\right)\)
3. \(\overrightarrow{\mathrm{E}}=\hat{\mathrm{i}} z^3+\hat{\mathrm{j}} x y z+\hat{k} z^2\)
4. \(\vec{E}=\hat{i}\left(2 x y-z^3\right)+\hat{j} x y^2+\hat{k} 3 z^2 x\)

Solution: 1. \(\vec{E}=\hat{i}\left(2 x y+z^3\right)+\hat{j} x^2+\hat{k} 3 x z^2\)

Question 2. The electric field at a distance \(\frac{3 R}{2}\)from the center of a charged conducting spherical shell of the radius is E. The electric field at a distance \(\frac{R}{2}\) from the centre of the sphere is

1. zero
2. E
3. E/2
4. E/3

Solution: 1. zero

Question 3. A charge Q is enclosed by a Gaussian spherical surface of radius R. If the radius is doubled, then the outward electric flux will :

1. Increase four times
2. Be reduced to half
3. Remain the same
4. Be doubled

Solution: 3. Remain the same

Question 4. Four electric charges +q, +q, –q, and –q are placed at the corners of a square of side 2L (see figure). The electric potential at point A, midway between the two charges +q and +q, is :

1. \(\frac{1}{4 \pi \epsilon_0} \frac{2 q}{L}(1+\sqrt{5})\)
2. \(\frac{1}{4 \pi \epsilon_0} \frac{2 \mathrm{q}}{\mathrm{L}}\left(1+\frac{1}{\sqrt{5}}\right)\)
3. \(\frac{1}{4 \pi \epsilon_0} \frac{2 q}{L}\left(1-\frac{1}{\sqrt{5}}\right)\)
4. Zero

Solution: 3. \(\frac{1}{4 \pi \epsilon_0} \frac{2 q}{L}\left(1-\frac{1}{\sqrt{5}}\right)\)

Question 5. The electric potential V at any point (x, y, z), all in meters in space is given by V = 4×2 volt. The electric field at the point (1, 0, 2) in volt/meter is :

1. 8 along the positive X-axis
2. 16 along the negative X-axis
3. 16 along the positive X-axis
4. 8 along the negative X-axis

Solution: 4. 8 along the negative X-axis

Question 6. Three charges, each +q, are placed at the corners of an isosceles triangle ABC of sides BC and AC, 2a. D and E are the midpoints of BC and CA. The work done in taking a charge Q from D to E is:

1. \(\frac{\mathrm{eqQ}}{8 \pi \epsilon_0 \mathrm{a}}\)
2. \(\frac{q Q}{4 \pi \epsilon_0 a}\)
3. zero
4. \(\frac{3 q Q}{4 \pi \epsilon_0 a}\)

Solution: 3. zero

Question 7. An electric dipole of the moment ´p´ is placed in an electric field of intensity ´E´. The dipole acquires a position such that the axis of the dipole makes an angle θ with the direction of the field. Assuming that the potential energy of the dipole is zero when θ = 90º, the torque and the potential energy of the dipole will respectively be :

1. p E sin θ, – p E cos θ
2. p E sin θ, – 2 p E cos θ
3. p E sin θ, 2 p Ecos θ
4. p E cos θ, – p Ecos θ

Solution: 1. p E sin θ, – p E cos θ

Question 8. Four point charges –Q, –q, 2q, and 2Q are placed, one at each corner of the square. The relation between Q and q for which the potential at the center of the square is zero is :

1. Q = – q
2. \(Q=-\frac{1}{q}\)
3. Q = q
4. \(Q=\frac{1}{q}\)

Solution: 1. Q = – q

Question 9. What is the flux through a cube of side ‘a’ if a point charge of q is at one of its corners:

1. \(\frac{2 \mathrm{q}}{\varepsilon_0}\)
2. \(\frac{q}{8 \varepsilon_0}\)
3. \(\frac{\mathrm{q}}{\varepsilon_0}\)
4. \(\frac{q}{2 \varepsilon_0} 6 a^2\)

Solution: 2. \(\frac{q}{8 \varepsilon_0}\)

Question 10. Two metallic spheres of radii 1 cm and 3 cm are given charges of –1×10^-2 C and 5×10^-2 C, respectively. If these are connected by a conducting wire, the final charge on the bigger sphere is :

1. 2×10^-2 C
2. 3×10^-2 C
3. 4×10^-2 C
4. 1×10^-2 C

Solution: 2. 3×10^-2 C

Question 11. A, B, and C are three points in a uniform electric field. The electric potential is :

1. Maximum at B
2. Maximum at C
3. Same at all three points A, B, and C
4. Maximum at A

Solution: 1. Maximum at B

Question 12. Two pith balls carrying equal charges are suspended from a common point by strings of equal length, the equilibrium separation between them is r. Now the strings are rigidly clamped at half the height. The equilibrium separation between the balls now becomes:

1. \(\left(\frac{r}{\sqrt[3]{2}}\right)\)
2. \(\left(\frac{2 r}{\sqrt{3}}\right)\)
3. \(\left(\frac{2 r}{3}\right)\)
4. \(\left(\frac{r}{\sqrt{2}}\right)^2\)

Solution: 1. \(\left(\frac{r}{\sqrt[3]{2}}\right)\)

Question 13. A conducting sphere of radius R is given a charge Q. The electric potential and the electric field at the center of the sphere respectively are:

1. \(\text { Zero } \& \frac{\mathrm{Q}}{4 \pi \varepsilon_0 \mathrm{R}^2}\)
2. \(\frac{\mathrm{Q}}{4 \pi \varepsilon_0 \mathrm{R}} \& \text { Zero }\)
3. \(\frac{\mathrm{Q}}{4 \pi \varepsilon_0 \mathrm{R}} \ \frac{\mathrm{Q}}{4 \pi \varepsilon_0 \mathrm{R}^2}\)
4. Both are zero.

Solution: 2. \(\frac{\mathrm{Q}}{4 \pi \varepsilon_0 \mathrm{R}} \& \text { Zero }\)

Question 14. In a region the potential is represented by V(x, y, z) = 6x – 8xy –8y + 6yz, where V is in volts and x, y, and z, are in meters. The electric force experienced by a charge of 2 coulomb situated at point (1, 1,1) is :

1. 6 √5N
2. 30N
3. 24N
4. 4 √35N

Solution: 4. 4 v35N

Question 15. The electric field in a certain region is acting radially outward and is given by E = Ar. A charge contained in a sphere of radius ‘a’ centered at the origin of the field, will given by :

1. Aa ∈a²
2. 4π∈0 Aa³
3. ∈0 Aa³
4. 4π∈0 Aa³

Solution: 2. 4π∈0 Aa³

Question 16. If potential (in volts) in a region is expressed as V(x, y, z) = 6 xy – y + 2yz, the electric field (in N/C) at point (1, 1, 0) is :

1. \(-(6 \hat{\mathbf{i}}+5 \hat{\mathbf{j}}+2 \hat{k})\)
2. \(-(2 \hat{i}+3 \hat{j}+\hat{k})\)
3. \(-(6 \hat{i}+9 \hat{j}+\hat{k})\)
4. \(-(3 \hat{\mathbf{i}}+5 \hat{\mathbf{j}}+3 \hat{\mathbf{k}})\)

Solution: 1. \(-(6 \hat{\mathbf{i}}+5 \hat{\mathbf{j}}+2 \hat{k})\)

Question 17. Two identical charged spheres suspended from a common point by two mass-less strings of lengths l are initially at a distance d(d << l) apart because of their mutual repulsion. The charges begin to leak from both spheres at a constant rate. As a result, the spheres approach each other with a velocity v. Then v varies as a function of the distance x between the spheres, as :

1. v ∝ x–1
2. v ∝ x1/2
3. v ∝ x
4. v ∝ x–1/2

Solution: 4. v ∝ x–1/2

Question 18. When an α-particle of mass ‘m’ moving with velocity ‘ v ‘ bombards on a heavy nucleus of charge ‘Ze’ its distance of closet approach from the nucleus depends on m as : 

1. m
2. \(\frac{1}{m}\)
3. \(\frac{1}{\sqrt{m}}\)
4. \(\frac{1}{\mathrm{~m}^2}\)

Solution: 2. \(\frac{1}{m}\)

Question 19. An electric dipole is placed at an angle of 30º with an electric field intensity of 2 ×105 N/C. It experiences a torque equal to 4 N m. The charge on the dipole, if the dipole length is 2cm, is

1. μC
2. 8 mC
3. 2 mC
4. 5 mC

Solution: 3. 2 mC

Question 20. Suppose the charge of a proton and an electron differ slightly. One of them is – e, and the other is (e + Δe). If the net of electrostatic force and the gravitational force between two hydrogen atoms placed at a distance d (much greater than atomic size) apart is zero, then Δe is of the order of [Given the mass of hydrogen mh = 1.67 × 10^-27 kg]

1. 10^-20 C
2. 10^-23 C
3. 10^-37 C
4. 10^-47 C

Solution: 3. 10–37 C

Question 21. The diagrams below show regions of equipotentials.

1. The positive charge is moved from A to B in each diagram
2. Maximum work is required to move q in Figure (c).
3. In all four cases, the work done is the same.
4. Minimum work is required to move q in Figure (a)
5. Maximum work is required to move q in Figure (b).

Solution: 2. In all four cases the work done is the same.

Question 22. An electron falls from rest through a vertical distance h in a uniform and vertically upward-directed electric field E. The direction of the electric field is now reversed, keeping its magnitude the same. A proton is allowed to fall from rest in it through the same vertical distance h. The time of fall of the electron, in comparison to the time of fall of the proton is : 

1. Smaller
2. Equal
3. 10 times greater
4. 5 times greater

Solution: 1. Smaller

Question 24. Two point charges A and B, having charges +Q and –Q respectively, are placed at a certain distance apart, and the force acting between them is F. If 25% charge of A is transferred to B, then the force between the charges becomes:

1. \(\frac{\mathrm{4F}}{\mathrm{3}}\)
2. F
3. \(\frac{\mathrm{9F}}{\mathrm{16}}\)
4. \(\frac{\mathrm{16F}}{\mathrm{9}}\)

Solution: 3. \(\frac{\mathrm{9F}}{\mathrm{16}}\)

Question 25. Two parallel infinite line charges with linear charge densities +λ C/m and –λ C/m are placed at a distance of 2R in free space. What is the electric field mid-way between the two line charges?

1. \(\frac{\lambda}{2 \pi \varepsilon_0 R} \mathrm{R} / \mathrm{C}\)
2. zero
3. \(\frac{2 \lambda}{\pi \varepsilon_0 R} N / C\)
4. \(\frac{\lambda}{\pi \varepsilon_0 R} N / C\)

Solution: 4. \(\frac{\lambda}{\pi \varepsilon_0 R} N / C\)

Question 26. Two metal spheres, one of radius R and the other of radius 2R respectively have the same surface charge density σ. They are brought in contact and separated. What will be the new surface charge densities on them?

1. \(\sigma_1=\frac{5}{6} \sigma, \sigma_2=\frac{5}{6} \sigma\)
2. \(\sigma_1=\frac{5}{2} \sigma, \sigma_2=\frac{5}{6} \sigma/\)
3. \(\sigma_1=\frac{5}{2} \sigma, \sigma_2=\frac{5}{3} \sigma\)
4. \(\sigma_1=\frac{5}{3} \sigma, \sigma_2=\frac{5}{6} \sigma\)

Solution: 4. \(\sigma_1=\frac{5}{3} \sigma, \sigma_2=\frac{5}{6} \sigma\)

Question 27. A sphere encloses an electric dipole with charges ±3 × 10^-6 C. What is the total electric flux across the sphere?

1. 3 × 10^-6
2. Zero
3. 3 × 10^-6 Nm²/C
4. 6 × 10^-6 Nm²/C

Solution: 2. Zero

Question 28. The electric field at a point on the equatorial plane at a distance r from the center of a dipole having dipole moment is given by (r >> separation of two charges forming the dipole,∈ −0permittivity of free space)

1. \(\overrightarrow{\mathrm{E}}=\frac{\overrightarrow{\mathrm{p}}}{4 \pi \epsilon_0 \mathrm{r}^3}\)
2. \(\overrightarrow{\mathrm{E}}=\frac{2 \overrightarrow{\mathrm{p}}}{4 \pi \epsilon_0 \mathrm{r}^3}\)
3. \(\vec{E}=-\frac{\vec{p}}{4 \pi \epsilon_0 r^2}\)
4. \(\overrightarrow{\mathrm{E}}=-\frac{\overrightarrow{\mathrm{p}}}{4 \pi \epsilon_0 \mathrm{r^3}}\)

Solution: 4. \(\overrightarrow{\mathrm{E}}=-\frac{\overrightarrow{\mathrm{p}}}{4 \pi \epsilon_0 \mathrm{r^3}}\)

Question 29. The acceleration of an electron due to the mutual attraction between the electron and a proton when they are Aapart is, \(\left(\mathrm{m}_{\mathrm{e}} \times 10^{-31} \mathrm{~kg}, \mathrm{e}=1.6 \times 10^{-19} \mathrm{C}\right)\left(\text { Take } \frac{1}{4 \pi \varepsilon_0}=9 \times 10^9 \mathrm{Nm}^2 \mathrm{C}^{-2}\right)\)

1. 10²⁴ m/s²
2. 10²³ m/s²
3. 10²² m/s²
4. 10²⁵ m/s²

Solution: 3. 1022 m/s²

Question 30. A spherical conductor of radius 10cm has a charge of 3.2 × 10^-7 C distributed uniformly. What is the magnitude of the electric field at a point 15 cm from the center of the sphere? \(\left(\frac{1}{4 \pi \varepsilon_0}=9 \times 10^9 \mathrm{Nm}^2 / \mathrm{C}^2\right)\)

1. 1.28 × 10⁷ N/C
2. 1.28 × 10⁴ N/C
3. 1.28 × 10⁵ N/C
4. 1.28 × 10⁶ N/C

Solution: 3. 1.28 × 10⁵ N/C

Question 31. Two points P and Q are maintained at the potentials of 10 V and –4 V respectively. The work done in moving 100 electrons from P to Q is :

1. 9.60 × 10^-17J
2. –2.24 × 10^-15 J
3. 2.24 × 10^-16 J
4. –9.60 × 10^-17 J

Solution: 3. 2.24 × 10^-16 J

Question 32. A charge Q is placed at each of the opposite corners of a square. A charge q is placed at each of the other two corners. If the net electrical force on Q is zero, then Q/q equals:

1. –1
2. 1
3. \(-\frac{1}{\sqrt{2}}\)
4. \(-2 \sqrt{2}\)

Solution: 4. \(-2 \sqrt{2}\)

Question 33. Statement 1: For a charged particle moving from point P to point Q, the net work done by an electrostatic field on the particle is independent of the path connecting point P to point Q.

Statement 2: The net work done by a conservative force on an object moving along a closed loop is zero.

1. Statement-1 is true, Statement-2 is true; Statement-2 is the correct explanation of Statement-1.
2. Statement-1 is true, Statement-2 is true; Statement-2 is not the correct explanation of Statement-1.
3. Statement-1 is false, Statement-2 is true.
4. Statement-1 is true, Statement-2 is false.

Solution: 1. Statement-1 is true, Statement-2 is true; Statement-2 is the correct explanation of Statement-1.

Question 34. A thin semi-circular ring of radius r has a positive charge q distributed uniformly over it. The net field E at the center O is

1. \(\frac{q}{4 \pi^2 \varepsilon_0 r^2} \hat{j}\)
2. \(-\frac{q}{4 \pi^2 \varepsilon_0 r^2} \hat{j}\)
3. \(-\frac{q}{2 \pi^2 \varepsilon_0 r^2} \hat{j}\)
4. \(\frac{q}{2 \pi^2 \varepsilon_0 r^2} \hat{j}\)

Solution: 3. \(-\frac{q}{2 \pi^2 \varepsilon_0 r^2} \hat{j}\)

Question 35. Two identical charged spheres are suspended by strings of equal lengths. The strings make an angle of 30º with each other. When suspended in a liquid of density 0.8 g cm–3, the angle remains the same. If the density of the material of the sphere is 1.6 g cm–3, the dielectric constant of the liquid is

1. 4
2. 3
3. 2
4. 1

Solution: 3. 2

Question 36. The electrostatic potential inside a charged spherical ball is given by φ = ar2 + b where r is the distance from the center; a,b are constants. Then the charge density inside the ball is :

1. –24π aε₀r
2. –6π aε₀r
3. –24π aε₀
4. –6 aε₀

Solution: 4. –6 aε₀

Question 37. Two identical charged spheres suspended from a common point by two massless strings of length l are initially a distance d(d < < l) apart because of their mutual repulsion. The charge begins to leak from both spheres at a constant rate. As a result, the charges approach each other with a velocity υ. Then as a function of distance x between them :

1. υ ∝ x–1/2
2. υ ∝ x–1
3. υ ∝ x1/2
4. υ ∝ x

Solution: 1. υ ∝ x–1/2

Question 38. Two positive charges of magnitude ‘q’ are placed at the ends of a side (side 1) of a square of side ‘2a’. Two negative charges of the same magnitude are kept at the other corners. Starting from rest, if a charge Q moves from the middle of side 1 to the center of the square, its kinetic energy at the center of the square is :

1. zero
2. \(\frac{1}{4 \pi \varepsilon_0} \frac{2 \mathrm{qQ}}{\mathrm{a}}\left(1+\frac{1}{\sqrt{5}}\right)\)
3. \(\frac{1}{4 \pi \varepsilon_0} \frac{2 q \mathrm{Q}}{\mathrm{a}}\left(1-\frac{2}{\sqrt{5}}\right)\)
4. \(\frac{1}{4 \pi \varepsilon_0} \frac{2 q Q}{a}\left(1-\frac{1}{\sqrt{5}}\right)\)

Solution: 4. \(\frac{1}{4 \pi \varepsilon_0} \frac{2 q Q}{a}\left(1-\frac{1}{\sqrt{5}}\right)\)

Question 39. In a uniformly charged sphere of total charge Q and radius R, the electric field E is plotted as a function of distance from the center. The graph that would correspond to the above will be :

Solution: 3.

Question 40. This question has statement 1 and statement 2. Of the four choices given after the statements, choose the one that best describes the two statements. An insulating solid sphere of radius R has a uniformly positive charge density ρ. As a result of this uniform charge distribution, there is a finite value of the electric potential at the center of the sphere, at the surface of the sphere, and also at a point outside the sphere. The electric potential at infinite is zero.

Statement-1: When a charge ‘q’ is taken from the center of the surface of the sphere its potential energy changes by \(\frac{\mathrm{q} \rho}{3 \varepsilon_0}\)

Statement-2: The electric field at a distance r (r < R) from the center of the sphere is \(\frac{\rho r}{3 \varepsilon_0}\)

1. Statement-1 is true, Statement-2 is true; Statement-2 is not the correct explanation of statement-1.
2. Statement 1 is true Statement 2 is false.
3. Statement 1 is false Statement 2 is true.
4. Statement 1 is true, Statement 2 is true, and Statement 2 is the correct explanation of Statement 1.

Solution: 3. Statement 1 is false Statement 2 is true.

Question 41. Two charges, each equal to q, are kept at x = – a and x = a on the x-axis. A particle of mass m and charge \(q_0=\frac{q}{2}\) is placed at the origin. If charge q0is given a small displacement (y <<a) along the y-axis, the net force acting on the particle is proportional to :

1. y
2. –y
3. \(\frac{1}{y}\)
4. \(-\frac{1}{y}\)

Solution: 1. y

Question 42. A charge Q is uniformly distributed over a long rod AB of length L as shown in the figure. The electric potential at the point O lying at distance L from the end A is :

1. \(\frac{Q}{8 \pi \in_0 L}\)
2. \(\frac{3 Q}{4 \pi \epsilon_0 L}\)
3. \(\frac{Q}{4 \pi \in_0 L \ln 2}\)
4. \(\frac{Q \ln 2}{4 \pi \epsilon_0 L}\)

Solution: 4. \(\frac{Q \ln 2}{4 \pi \epsilon_0 L}\)

Question 43. Assume that an electric field \(\overrightarrow{\mathrm{E}}=30 \mathrm{x}^2 \hat{\mathrm{i}}\) exists in space. Then the potential difference VA– VO, where VO is the potential at the origin and the potential at x = 2 m is :

1. 120 V
2. –120 V
3. – 80 V
4. 80 V

Solution: 3. – 80 V

Question 44. A long cylindrical shell carries a positive surface charge σ in the upper half and a negative surface charge – σ in the lower half. The electric field lines around the cylinder will look like the figure given in : (figures are schematic and not drawn to scale)

Solution: 1.

Question 45. A uniformly charged solid sphere of radius R has potential V0(measured concerning ∞) on its surface. For this sphere the equipotential surfaces with potentials \(\frac{3 \mathrm{~V}_0}{2}, \frac{5 \mathrm{~V}_0}{4}, \frac{3 \mathrm{~V}_0}{4} \text { and } \frac{\mathrm{V}_0}{4}\) have radius R₁, R2, R3and R4respectively. Then

1. R₁= 0 and R₂> (R₄– R₃)
2. R₁ ≠0 and (R₂– R₁) > (R₄– R₃)
3. R₁= 0 and R₂< (R₄– R₃)
4. 2R < R₄

Solution: 3. R₁= 0 and R₂< (R₄– R₃)

Question 46. The region between two concentric spheres of radii ‘a’ and ‘b’, respectively (see figure), has volume charge density Ar ρ=\(\), where A is a constant and r is the distance from the center. At the center of the spheres is a point charge Q. The value of A such that the electric field in the region between the spheres will be constant, is :

1. \(\frac{Q}{2 \pi\left(b^2-a^2\right)}\)
2. \(\frac{2 Q}{\pi\left(a^2-b^2\right)}\)
3. \(\frac{2 Q}{\pi a^2}\)
4. \(\frac{Q}{2 \pi a^2}\)

Solution: 4. \(\frac{Q}{2 \pi a^2}\)

Question 47. An electric dipole has a fixed dipole moment \(\overrightarrow{\mathrm{p}}\), which makes angle θ concerning the x-axis. When subjected to an electric field \(\overrightarrow{\mathrm{E}}_1=\mathrm{E} \hat{\mathbf{i}},\), it experiences a torque \(\overrightarrow{\mathrm{T}}_1=\tau \hat{\mathrm{k}}\). When subjected to another \(\overrightarrow{\mathrm{E}}_2=\sqrt{3} \mathrm{E}_1 \hat{\mathrm{j}}\) electric field ˆit experiences a torque \(\overrightarrow{\mathrm{T}}_2=-\overrightarrow{\mathrm{T}}_1\). The angle θ is :

1. 90°
2. 30°
3. 45°
4. 60°

Solution: 4. 60°

Question 48. Three concentric metal shells A, B, and C of respective radii a,b, and c (a < b < c) have surface charge densities +σ, –σ, and +σ respectively. The potential of shell B is :

1. \(\frac{\sigma}{\varepsilon_0}\left[\frac{b^2-c^2}{b}+a\right]\)
2. \(\frac{\sigma}{\varepsilon_0}\left[\frac{\mathrm{b}^2-\mathrm{c}^2}{\mathrm{c}}+\mathrm{a}\right]\)
3. \(\frac{\sigma}{\varepsilon_0}\left[\frac{\mathrm{a}^2-\mathrm{b}^2}{\mathrm{a}}+\mathrm{c}\right]\)
4. \(\frac{\sigma}{\varepsilon_0}\left[\frac{\mathrm{a}^2-\mathrm{b}^2}{\mathrm{~b}}+\mathrm{c}\right]\)

Solution: 4. \(\frac{\sigma}{\varepsilon_0}\left[\frac{\mathrm{a}^2-\mathrm{b}^2}{\mathrm{~b}}+\mathrm{c}\right]\)

Question 49. Three charges +Q, q, +Q are placed respectively, at distance, 0, d/2, and d from the origin, on the x-axis. If the net force experienced by +Q, placed at x = 0, is zero, then the value of q is :

1. +Q/2
2. +Q/4
3. –Q/2
4. –Q/4

Solution: 4. –Q/4

Question 50. For a uniformly charged ring of radius R, the electric field on its axis has the largest magnitude at a distance of h from its center. The value of h is :

1. \(\frac{\mathrm{R}}{\sqrt{2}}\)
2. R
3. \(\frac{\mathrm{R}}{\sqrt{5}}\)
4. R 2

Solution: 1. \(\frac{\mathrm{R}}{\sqrt{2}}\)

Question 51. Two point charges \(\rho(r)=\frac{A}{r^2} e^{-2 r / a}\)and q2( −25μCa )re placed on the x-axis at x = 1 m and x = 4 m respectively. The electric field (in V/m) at a point y = 3 m on y-axis is,\(\left[\text { take }=\frac{1}{4 \pi \mathrm{g} \varepsilon_0}=9 \times 10^9 \mathrm{Nm}^2 \mathrm{C}^{-2}\right]\)

1. \((-81 \hat{i}+81 \hat{j}) \times 10^2\)
2. \((81 \hat{\mathrm{i}}-81 \hat{\mathrm{j}}) \times 10^2\)
3. \((-63 \hat{\mathbf{i}}+27 \hat{\mathbf{j}}) \times 10^2\)
4. \((63 \hat{\mathbf{i}}-27 \hat{\mathrm{j}}) \times 10^2\)

Solution: 4. \((63 \hat{\mathbf{i}}-27 \hat{\mathrm{j}}) \times 10^2\)

Question 52. Charge is distributed within a sphere of radius R with a volume charge density \(\), where A 2e r and a are constant. If Q is the total charge of this charge distribution, the radius R is :

1. \(\frac{Q}{12 \pi \epsilon_0} \frac{a b+b c+c a}{a b c}\)
2. \(\frac{a}{2} \log \left(\frac{1}{1-\frac{Q}{2 \pi a A}}\right)\)
3. \(\frac{a}{2} \log \left(1-\frac{Q}{2 \pi a A}\right)\)
4. \(\mathrm{a} \log \left(1-\frac{\mathrm{Q}}{2 \pi \mathrm{aA}}\right)\)

Solution: 2. \(\frac{Q}{12 \pi \epsilon_0} \frac{a b+b c+c a}{a b c}\)

Question 53. A charge Q is distributed over three concentric spherical shells of radii a, b, c (a < b < c) such that their surface charge densities are equal. The total potential at a point at distance r from their common center, where r < a, would be

1. \(a \log \left(\frac{1}{1-\frac{Q}{2 \pi a A}}\right)\)
2. \(\frac{Q}{4 \pi \in_0(a+b+c)} \)
3. \(\frac{Q(a+b+c)}{4 \pi \epsilon_0\left(a^2+b^2+c^2\right)}\)
4. \(\frac{Q\left(a^2+b^2+c^2\right)}{4 \pi \epsilon_0\left(a^3+b^3+c^3\right)}\)

Solution: 3. \(a \log \left(\frac{1}{1-\frac{Q}{2 \pi a A}}\right)\)

Question 54. Two electric dipoles, A, B with respective dipole moments A d \(\overrightarrow{\mathrm{d}}_{\mathrm{A}}=-4 q a \hat{i} \text { and } \overrightarrow{\mathrm{d}}_{\mathrm{B}}=-2 q a \hat{i}\) is placed on x-axis with a separation R, as shown in the figure.

The distance from A at which both of them produce the same potential is :

1. \(\frac{R}{\sqrt{2}-1}\)
2. \(\frac{\sqrt{2} R}{\sqrt{2}+1}\)
3. \(\frac{\sqrt{2} R}{\sqrt{2}-1}\)
4. \(\frac{R}{\sqrt{2}+1}\)

Solution: 3. \(\frac{\sqrt{2} R}{\sqrt{2}-1}/\)

Question 55. Four equal point charges Q each are placed in the xy-plane at (0, 2), (4, 2), (4, –2), and (0, –2). The work required to put a fifth charge Q at the origin of the coordinate system will be

1. \(\frac{Q^2}{4 \pi \in_0}\left(1+\frac{1}{\sqrt{3}}\right)\)
2. \(\frac{Q^2}{4 \pi \epsilon_0}\left(1+\frac{1}{\sqrt{5}}\right)\)
3. \(\frac{Q^2}{4 \pi \epsilon_0}\)
4. \(\frac{Q^2}{2 \sqrt{2} \pi \epsilon_0}\)

Solution: 2. \(\frac{Q^2}{4 \pi \epsilon_0}\left(1+\frac{1}{\sqrt{5}}\right)\)

Question 56. Charge –q and +q located at A and B, respectively, constitute an electric dipole. The distance AB = 2a, O is the mid-point of the dipole and OP is perpendicular to AB. A charge Q is placed at P where y and y >> 2a. The charge Q experiences an electrostatic force F. If Q is now moved along the equatorial line y, the force on Q will be close to : \(\left(\frac{y}{3}>2 a\right)\)

1. 3F
2. 27F
3. 9F
4. F/3

Solution: 2. 27F

Question 57. The given graph shows a variation (with distance r from center) of :

1. The electric field of a uniformly charged sphere
2. Potential of a uniformly charged spherical shell
3. The potential of a uniformly charged sphere
4. The electric field of a uniformly charged spherical shell

Solution: 2. Potential of a uniformly charged spherical shell

Question 58. Three charges Q, +q, and +q and placed at the vertices of a right-angle isosceles triangle as shown below. The net electrostatic energy of the configuration is zero, if the value of Q is :

1. +q
2. \(\frac{-\sqrt{2} q}{\sqrt{2}+1}\)
3. -2q
4. \(\frac{-q}{1+\sqrt{2}}\)

Solution: 2. \(\frac{-\sqrt{2} q}{\sqrt{2}+1}\)

Question 59. A particle of mass m and charge q is in an electric and magnetic field given by \(\overrightarrow{\mathrm{E}}=2 \hat{i}+3 \hat{j}; \vec{B}=4 \hat{j}+6 \hat{k}\) The charged particle is shifted from the origin to the point P(x = 1; y = 1) along a straight path. The magnitude of the total work done is :

1. 5q
2. (2.5)q
3. (0.35) q
4. (0.15)q

Solution: 1. 5q

Question 60. An electric field of 1000 V/m is applied to an electric dipole at an angle of 45°. The value of the electric dipole moment is 10–29 C.m. What is the potential energy of the electric dipole?

1. –9 × 10^-20 J
2. –7 × 10^-27 J
3. –10 × 10^-29 J
4. –20 × 10^-18 J

Solution: 2. –7 × 10^-27 J

Question 61. Determine the electric dipole moment of the system of three charges, placed on the vertices of an equilateral triangle, as shown in the figure :

1. \(-\sqrt{3} q \ell \hat{j}\)
2. \(\left(q \ell \ell \frac{\hat{i}+\hat{j}}{\sqrt{2}}\right.\)
3. \(2 q \ell \hat{j}\)
4. \(\sqrt{3} \mathrm{q} \ell \frac{\hat{\mathrm{j}}-\hat{\mathrm{i}}}{\sqrt{2}}\)

Solution: 1. \(-\sqrt{3} q \ell \hat{j}\)

Question 62. There is a uniform spherically symmetric surface charge density at a distance R0 from the origin. The charge distribution is initially at rest and starts expanding because of mutual repulsion. The figure that represents best the speed V(R(t)) of the distribution as a function of its instantaneous radius R(t) is:

Solution: 4.

Question 63. A sphere of radius 1 cm has a potential of 8000 V. The energy density near the surface of the sphere will be:

1. 64 × 105 J/m³
2. 8 × 103 J/m³
3. 2 J/m³
4. 2.83 J/m³

Solution: 4. 2.83 J/m³

Question 64. In the above question, the electric force acting on a point charge of 2 C placed at the origin will be :

1. 2 N
2. 500 N
3. –5 N
4. –500 N

Solution: 4. –500 N

Question 65. The figure shows two large cylindrical shells having uniform linear charge densities + λ and – λ. The radius of the inner cylinder is ‘a’ and that of the outer cylinder is ‘b’. A charged particle of mass m, charge q revolves in a circle of radius r. Then, its speed ‘v’ is : (Neglect gravity and assume the radii of both the cylinders to be very small in comparison to their length.)

1. \(\sqrt{\frac{\lambda q}{2 \pi \epsilon_0 m}}\)
2. \(\sqrt{\frac{2 \lambda q}{\pi \epsilon_0 m}}\)
3. \(\sqrt{\frac{\lambda q}{\pi \epsilon_0 m}}\)
4. \(\sqrt{\frac{\lambda \mathrm{q}}{4 \pi \varepsilon_0 \mathrm{~m}}}\)

Solution: 1. \(\sqrt{\frac{\lambda q}{2 \pi \epsilon_0 m}}\)

Question 66. A charge q is uniformly distributed over a large plastic plate. The electric field at point P close to the center and just above the surface of the plate is 50 V/m. If the plastic plate is replaced by a copper plate of the same geometrical dimensions and carrying the same uniform charge q, the electric field at point P will become:

1. zero
2. 25 V/m
3. 50 V/m
4. 100 V/m

Solution: 3. 50 V/m

Question 67. A point charge q is brought from infinity (slowly so that heat developed in the shell is negligible) and is placed at the center of a conducting neutral spherical shell of inner radius a and outer radius b, then work done by the external agent is:

1. 0
2. \(\frac{k q^2}{2 b}\)
3. \(\frac{k q^2}{2 b}-\frac{k q^2}{2 a}\)
4. \(\frac{k q^2}{2 a}-\frac{k q^2}{2 b}\)

Solution: 3. \(\frac{k q^2}{2 b}-\frac{k q^2}{2 a}\)

Question 68. The magnitude of the electric force on 2 μ c charge placed at the center O of two equilateral triangles each of side 10 cm, as shown in the figure is P. If charge A, B, C, D, E, and F are 2 μ c, 2 μ c, 2 μ c, -2 μc, – 2 μ c, – 2μ c respectively, then P is:

1. 21.6 N
2. 64.8 N
3. 0
4. 43.2 N

Solution: 4. 43.2 N

Question 69. A tiny spherical oil drop carrying a net charge q is balanced in still air with a vertical uniform electric field 81 5 of strength \(\frac{81 \pi}{7} \times 10^5 \mathrm{Vm}^{-1}\). When the field is switched off, the drop is observed to fall with terminal velocity 2 × 10–3 m s–1. Given g = 9.8 m s^-2, the viscosity of the air = 1.8 × 10–5 Ns m^-2, and the density of oil = 900 kg m–3, the magnitude of q is :

1. 1.6 × 10^-19 C
2. 3.2 × 10^-19 C
3. 4.8 × 10^-19 C
4. 8.0 × 10^-19 C

Solution: 4. 8.0 × 10^-19 C

Question 70. Identical charges (–q) are placed at each corner of a cube of side b, then the electrostatic potential energy of charge (+q) placed at the center of the cube will be :

1. \(-\frac{4 \sqrt{2} q^2}{\pi \varepsilon_0}\)
2. \(\frac{8 \sqrt{2} q^2}{\pi \varepsilon_0 \mathrm{~b}}\)
3. \(-\frac{4 q^2}{\sqrt{3} \pi \varepsilon_0 b}\)
4. \(\frac{8 \sqrt{2} q^2}{4 \pi \varepsilon_0 b}\)

Solution: 3. \(-\frac{4 q^2}{\sqrt{3} \pi \varepsilon_0b}\)

Question 71. Three charges Q, + q, and + q are placed at the vertices of a right-angled isosceles triangle as shown. The net electrostatic energy of the configuration is zero if Q is equal to :

1. \(\frac{-q}{1+\sqrt{2}}\)
2. \(\frac{-2 q}{2+\sqrt{2}}\)
3. -2q
4. +q

Solution: 2. \(\frac{-2 q}{2+\sqrt{2}}\)

Question 72. Six-point charges are kept at the vertices of a regular hexagon of side L and center O, as shown in the 1 q K figure. Given that \(\mathrm{K}=\frac{1}{4 \pi \varepsilon_0} \frac{\mathrm{q}}{\mathrm{L}^2}\), which of the following statement (s) is incorrect?

1. The electric field at O is 6K along OD
2. The potential at O is zero
3. The potential at all points on the line PR is the same
4. The potential at all points on the line ST is the same.

Solution: 4. The potential at all points on the line ST is the same.

Question 73. Two non-conducting solid spheres of radii R and 2R, having uniform volume charge densities ρ₁and ρ₂ respectively, touch each other. The net electric field at a distance 2R from the center of the smaller ρ₁ sphere, along the line joining the centers of the spheres, is zero. The ratio ρcan be ;

1. –4
2. 2
3. 32/25
4. 4

Solution: 4. 4

Question 74. Let E₁(r), E₂(r), and E₃(r) be the respective electric fields at a distance r from a point charge Q, an infinitely long wire with constant linear charge density λ, and an infinite plane with uniform surface charge density σ. if E₁(r0) = E₂(r0) = E₃(r0) at a given distance r0, then

1. \(\mathrm{Q}=4 \sigma \pi \mathrm{r}_0^2\)
2. \(r_0=\frac{\lambda}{2 \pi \sigma}\)
3. \(E_1\left(r_0 / 2\right)=2 E_2\left(r_0 / 2\right)\)
4. \(E_2\left(r_0 / 2\right)=4 E_3\left(r_0 / 2\right)\)

Solution: 3. \(E_1\left(r_0 / 2\right)=2 E_2\left(r_0 / 2\right)\)