Electric Potential Problems

This section provides 100 problems to test your understanding of electric potential, including potential calculations for point charges and charge distributions, potential energy, field-potential relations, and applications in conductors and capacitors. Inspired by JEE Main, JEE Advanced, and NEET exam patterns, these problems are tailored for exam preparation, offering a mix of numerical, conceptual, and derivation-based challenges. NEET-style problems (66–100) are formatted as multiple-choice questions (MCQs) to match the exam’s objective format. Problems are organized by type to support progressive learning and build confidence in mastering electrostatics, a key topic for JEE/NEET success.

Numerical Problems

1. Calculate the electric potential at a distance $r = 0.2 , \text{m}$ from a point charge $Q = 5 , \mu\text{C}$ ($k = 9 \times 10^9 , \text{N·m}^2/\text{C}^2$).

   • (a) $2.24 \times 10^5 , \text{V}$
   • (b) $2.25 \times 10^5 , \text{V}$
   • (c) $2.26 \times 10^5 , \text{V}$
   • (d) $2.27 \times 10^5 , \text{V}$

2. Two charges $q_1 = +3 , \mu\text{C}$ at $(0, 0)$ and $q_2 = -3 , \mu\text{C}$ at $(0.4, 0)$. Calculate the electric potential at $(0.2, 0)$.

   • (a) $0 , \text{V}$
   • (b) $1 \times 10^5 , \text{V}$
   • (c) $2 \times 10^5 , \text{V}$
   • (d) $3 \times 10^5 , \text{V}$

3. Calculate the potential energy of two charges $q_1 = 4 , \mu\text{C}$ and $q_2 = -2 , \mu\text{C}$ separated by $r = 0.1 , \text{m}$.

   • (a) $-0.719 , \text{J}$
   • (b) $-0.720 , \text{J}$
   • (c) $-0.721 , \text{J}$
   • (d) $-0.722 , \text{J}$

4. A charge $q = 1 , \mu\text{C}$ moves between points with potentials $V_a = 200 , \text{V}$ and $V_b = 100 , \text{V}$. Calculate the work done by the electric field.

   • (a) $9.9 \times 10^{-5} , \text{J}$
   • (b) $1.00 \times 10^{-4} , \text{J}$
   • (c) $1.01 \times 10^{-4} , \text{J}$
   • (d) $1.02 \times 10^{-4} , \text{J}$

5. A line charge with $\lambda = 2 \times 10^{-6} , \text{C/m}$, length $L = 0.5 , \text{m}$, lies along the x-axis from $-0.25$ to $0.25$. Calculate the potential at $(0, 0.3)$ (approximate).

   • (a) $2.39 \times 10^4 , \text{V}$
   • (b) $2.40 \times 10^4 , \text{V}$
   • (c) $2.41 \times 10^4 , \text{V}$
   • (d) $2.42 \times 10^4 , \text{V}$

6. A ring of charge (radius $R = 0.1 , \text{m}$, total charge $Q = 4 , \mu\text{C}$) lies in the xy-plane. Calculate the potential on the z-axis at $z = 0.1 , \text{m}$.

   • (a) $2.54 \times 10^5 , \text{V}$
   • (b) $2.55 \times 10^5 , \text{V}$
   • (c) $2.56 \times 10^5 , \text{V}$
   • (d) $2.57 \times 10^5 , \text{V}$

7. A disk of radius $R = 0.2 , \text{m}$, surface charge density $\sigma = 3 \times 10^{-6} , \text{C/m}^2$, lies in the xy-plane. Calculate the potential on the z-axis at $z = 0.1 , \text{m}$ ($\epsilon_0 = 8.85 \times 10^{-12} , \text{C}^2/\text{N·m}^2$).

   • (a) $6.27 \times 10^4 , \text{V}$
   • (b) $6.28 \times 10^4 , \text{V}$
   • (c) $6.29 \times 10^4 , \text{V}$
   • (d) $6.30 \times 10^4 , \text{V}$

8. A spherical shell of radius $R = 0.1 , \text{m}$ has total charge $Q = 6 , \mu\text{C}$. Calculate the potential at $r = 0.05 , \text{m}$.

   • (a) $5.39 \times 10^5 , \text{V}$
   • (b) $5.40 \times 10^5 , \text{V}$
   • (c) $5.41 \times 10^5 , \text{V}$
   • (d) $5.42 \times 10^5 , \text{V}$

9. A potential varies as $V = k \frac{3 \times 10^{-6}}{r}$. Calculate the electric field at $r = 0.3 , \text{m}$.

   • (a) $9.99 \times 10^4 , \text{N/C}$
   • (b) $1.00 \times 10^5 , \text{N/C}$
   • (c) $1.01 \times 10^5 , \text{N/C}$
   • (d) $1.02 \times 10^5 , \text{N/C}$

10. A uniform field $E = 400 \hat{i} , \text{N/C}$ exists between $(0, 0)$ and $(0.2, 0)$. Calculate the potential difference $V_{(0,0)} - V_{(0.2,0)}$.

    • (a) $79.9 , \text{V}$
    • (b) $80.0 , \text{V}$
    • (c) $80.1 , \text{V}$
    • (d) $80.2 , \text{V}$

11. A spherical conductor of radius $R = 0.2 , \text{m}$ has charge $Q = 8 , \mu\text{C}$. Calculate the potential on the surface.

    • (a) $3.59 \times 10^5 , \text{V}$
    • (b) $3.60 \times 10^5 , \text{V}$
    • (c) $3.61 \times 10^5 , \text{V}$
    • (d) $3.62 \times 10^5 , \text{V}$

12. A parallel plate capacitor has area $A = 0.01 , \text{m}^2$, separation $d = 0.001 , \text{m}$, and charge $Q = 1 \times 10^{-8} , \text{C}$. Calculate the potential difference across the plates.

    • (a) $112.9 , \text{V}$
    • (b) $113.0 , \text{V}$
    • (c) $113.1 , \text{V}$
    • (d) $113.2 , \text{V}$

13. A capacitor with capacitance $C = 20 , \mu\text{F}$ is charged to $V = 50 , \text{V}$. Calculate the energy stored in the capacitor.

    • (a) $2.49 \times 10^{-2} , \text{J}$
    • (b) $2.50 \times 10^{-2} , \text{J}$
    • (c) $2.51 \times 10^{-2} , \text{J}$
    • (d) $2.52 \times 10^{-2} , \text{J}$

14. Three charges $q_1 = q_2 = q_3 = 2 , \mu\text{C}$ are at the vertices of an equilateral triangle with side $0.2 , \text{m}$. Calculate the total potential energy of the system.

    • (a) $0.539 , \text{J}$
    • (b) $0.540 , \text{J}$
    • (c) $0.541 , \text{J}$
    • (d) $0.542 , \text{J}$

15. Calculate the potential at $r = 0.5 , \text{m}$ from a point charge $Q = -7 , \mu\text{C}$.

    • (a) $-1.259 \times 10^5 , \text{V}$
    • (b) $-1.260 \times 10^5 , \text{V}$
    • (c) $-1.261 \times 10^5 , \text{V}$
    • (d) $-1.262 \times 10^5 , \text{V}$

16. Two charges $q_1 = +5 , \mu\text{C}$ at $(0.1, 0)$ and $q_2 = -5 , \mu\text{C}$ at $(-0.1, 0)$. Calculate the potential at $(0, 0.2)$.

    • (a) $0 , \text{V}$
    • (b) $1 \times 10^5 , \text{V}$
    • (c) $2 \times 10^5 , \text{V}$
    • (d) $3 \times 10^5 , \text{V}$

17. Calculate the potential energy of $q_1 = 6 , \mu\text{C}$ and $q_2 = 3 , \mu\text{C}$ separated by $r = 0.3 , \text{m}$.

    • (a) $0.539 , \text{J}$
    • (b) $0.540 , \text{J}$
    • (c) $0.541 , \text{J}$
    • (d) $0.542 , \text{J}$

18. A line charge with $\lambda = 1 \times 10^{-6} , \text{C/m}$, length $L = 0.6 , \text{m}$, lies along the x-axis. Calculate the potential at $(0, 0.1)$ (approximate).

    • (a) $1.79 \times 10^4 , \text{V}$
    • (b) $1.80 \times 10^4 , \text{V}$
    • (c) $1.81 \times 10^4 , \text{V}$
    • (d) $1.82 \times 10^4 , \text{V}$

19. A ring of charge (radius $R = 0.2 , \text{m}$, $Q = 5 , \mu\text{C}$) lies in the xy-plane. Calculate the potential at $z = 0.2 , \text{m}$.

    • (a) $1.59 \times 10^5 , \text{V}$
    • (b) $1.60 \times 10^5 , \text{V}$
    • (c) $1.61 \times 10^5 , \text{V}$
    • (d) $1.62 \times 10^5 , \text{V}$

20. A disk of radius $R = 0.1 , \text{m}$, $\sigma = 4 \times 10^{-6} , \text{C/m}^2$, lies in the xy-plane. Calculate the potential at $z = 0.05 , \text{m}$.

    • (a) $8.47 \times 10^4 , \text{V}$
    • (b) $8.48 \times 10^4 , \text{V}$
    • (c) $8.49 \times 10^4 , \text{V}$
    • (d) $8.50 \times 10^4 , \text{V}$

21. A spherical shell of radius $R = 0.3 , \text{m}$ has $Q = 10 , \mu\text{C}$. Calculate the potential at $r = 0.2 , \text{m}$.

    • (a) $2.99 \times 10^5 , \text{V}$
    • (b) $3.00 \times 10^5 , \text{V}$
    • (c) $3.01 \times 10^5 , \text{V}$
    • (d) $3.02 \times 10^5 , \text{V}$

22. A potential varies as $V = 1000 - 300x , \text{V}$. Calculate the electric field.

    • (a) $299 \hat{i} , \text{N/C}$
    • (b) $300 \hat{i} , \text{N/C}$
    • (c) $301 \hat{i} , \text{N/C}$
    • (d) $302 \hat{i} , \text{N/C}$

23. A spherical conductor of radius $R = 0.5 , \text{m}$ has $Q = 15 , \mu\text{C}$. Calculate the potential on the surface.

    • (a) $2.69 \times 10^5 , \text{V}$
    • (b) $2.70 \times 10^5 , \text{V}$
    • (c) $2.71 \times 10^5 , \text{V}$
    • (d) $2.72# Derivation Solutions

Derivation Problems

36. Derive the electric potential due to a point charge $V = k \frac{Q}{r}$.

37. Derive the potential energy of a system of two point charges $U = k \frac{q_1 q_2}{r}$.

38. Derive the work done by the electric field $W = q (V_a - V_b)$.

39. Derive the potential due to a uniform line charge at a perpendicular distance.

40. Derive the potential due to a ring of charge on its axis.

41. Derive the potential due to a uniformly charged disk on its axis.

42. Derive the potential inside and outside a spherical shell.

43. Derive the electric field from a potential $V = k \frac{Q}{r}$.

44. Derive the potential difference between two points in a uniform electric field.

45. Derive the potential on the surface of a spherical conductor.

46. Derive the potential difference across a parallel plate capacitor $V = \frac{Q d}{\epsilon_0 A}$.

47. Derive the energy stored in a capacitor $U = \frac{1}{2} C V^2$.

48. Derive the total potential energy of a system of three charges at the vertices of a triangle.

49. Derive the electric field from a potential $V = ax + by + cz$.

50. Derive the relation between equipotential surfaces and electric field lines.

NEET-style Conceptual Problems

66. What is the unit of electric potential in SI units?

• (a) Volt
• (b) Joule
• (c) Newton/Coulomb
• (d) Watt

67. What does a negative potential energy between two charges indicate?

• (a) Repulsive force
• (b) Attractive force
• (c) No force
• (d) Perpendicular force

68. What does the principle of superposition state for electric potential?

• (a) Potentials are vectors
• (b) Potentials add as scalars
• (c) Potentials cancel out
• (d) Potentials are independent of distance

69. What happens to the potential if the distance from a point charge doubles?

• (a) Doubles
• (b) Halves
• (c) Quadruples
• (d) Quarters

70. What is the dimension of electric potential?

• (a) $[\text{M} \text{L}^2 \text{T}^{-3} \text{A}^{-1}]$
• (b) $[\text{M} \text{L} \text{T}^{-1}]$
• (c) $[\text{L} \text{T}^{-2}]$
• (d) $[\text{M} \text{L}^2 \text{T}^{-1}]$

71. What does the electric field direction indicate relative to equipotential surfaces?

• (a) Parallel
• (b) Perpendicular
• (c) Random
• (d) No relation

72. What is the role of integration in potential calculations for charge distributions?

• (a) Sums vector potentials
• (b) Sums scalar potentials
• (c) Reduces potential
• (d) Increases distance

73. What happens to the potential inside a spherical shell?

• (a) Increases with radius
• (b) Decreases with radius
• (c) Constant
• (d) Zero

74. Why is the potential inside a conductor constant?

• (a) Due to high charge density
• (b) Due to $E = 0$ inside
• (c) Due to field lines
• (d) Due to charge quantization

75. What is the unit of capacitance in SI units?

• (a) Farad
• (b) Volt
• (c) Joule
• (d) Ohm

76. What does a constant potential inside a conductor indicate?

• (a) Non-zero electric field
• (b) Zero electric field
• (c) Variable field
• (d) Infinite field

77. Which type of surface is perpendicular to the electric field?

• (a) Equipotential surface
• (b) Field line surface
• (c) Charged surface
• (d) Conductor surface

78. What is the direction of the potential gradient?

• (a) Along the field
• (b) Opposite to the field
• (c) Perpendicular to the field
• (d) Random

79. What does a pseudo-force do in a non-inertial frame for potential calculations?

• (a) Affects perceived potential
• (b) Affects charge distribution
• (c) Creates field lines
• (d) Reduces potential

80. What is the dimension of potential energy?

• (a) $[\text{M} \text{L}^2 \text{T}^{-2}]$
• (b) $[\text{M} \text{L} \text{T}^{-1}]$
• (c) $[\text{L} \text{T}^{-2}]$
• (d) $[\text{M} \text{L}^2 \text{T}^{-1}]$

81. What is the role of electric potential in rocket ion propulsion?

• (a) Reduces charge
• (b) Determines ion energy for thrust
• (c) Increases distance
• (d) Decreases field

82. What happens to the potential inside a conductor with a cavity containing no charge?

• (a) Varies
• (b) Constant
• (c) Zero
• (d) Infinite

83. Why does the potential due to a point charge follow a $1/r$ dependence?

• (a) Due to symmetry
• (b) Due to integration of the field
• (c) Due to field lines
• (d) Due to charge quantization

84. What is the significance of $\frac{Q d}{\epsilon_0 A}$?

• (a) Potential difference across a capacitor
• (b) Electric field in a capacitor
• (c) Energy stored in a capacitor
• (d) Charge on a capacitor

85. What is the unit of energy stored in a capacitor?

• (a) Joule
• (b) Volt
• (c) Farad
• (d) Watt

86. What does a zero potential difference between two points indicate?

• (a) No electric field
• (b) Same potential
• (c) Maximum field
• (d) No charge

87. What is the physical significance of $k \int \frac{dq}{r}$?

• (a) Electric field
• (b) Potential due to a charge distribution
• (c) Potential energy
• (d) Charge density

88. Why does the electric field point from higher to lower potential?

• (a) Due to $\vec{E} = -\nabla V$
• (b) Due to symmetry
• (c) Due to field lines
• (d) Due to charge quantization

89. What is the dimension of capacitance?

• (a) $[\text{M}^{-1} \text{L}^{-2} \text{T}^4 \text{A}^2]$
• (b) $[\text{M} \text{L} \text{T}^{-1}]$
• (c) $[\text{L} \text{T}^{-2}]$
• (d) $[\text{M} \text{L}^2 \text{T}^{-1}]$

90. How does potential analysis help in ion propulsion systems?

• (a) Increases charge
• (b) Determines energy for ion acceleration
• (c) Reduces field
• (d) Increases distance

91. What is the role of distance in potential calculations?

• (a) Linear dependence
• (b) Inverse dependence
• (c) No dependence
• (d) Exponential dependence

92. What does a high potential difference in a capacitor indicate?

• (a) Low energy stored
• (b) High energy stored
• (c) No energy stored
• (d) Constant energy

93. What is the physical significance of $-\nabla V$?

• (a) Potential energy
• (b) Electric field
• (c) Charge density
• (d) Potential difference

94. What is the dimension of $\vec{E} \cdot \vec{l}$ in potential difference calculations?

• (a) $[\text{M} \text{L}^2 \text{T}^{-3} \text{A}^{-1}]$
• (b) $[\text{M} \text{L} \text{T}^{-1}]$
• (c) $[\text{L} \text{T}^{-2}]$
• (d) $[\text{M} \text{L}^2 \text{T}^{-3} \text{A}^{-1}]$

95. Why does the potential energy of like charges increase with decreasing distance?

• (a) Due to repulsive force
• (b) Due to attractive force
• (c) Due to field lines
• (d) Due to charge quantization

NEET-style Numerical Problems

96. Calculate the potential at $r = 0.4 , \text{m}$ from a point charge $Q = 2 , \mu\text{C}$.

• (a) $4.49 \times 10^4 , \text{V}$
• (b) $4.50 \times 10^4 , \text{V}$
• (c) $4.51 \times 10^4 , \text{V}$
• (d) $4.52 \times 10^4 , \text{V}$

97. A ring of charge (radius $R = 0.1 , \text{m}$, $Q = 3 , \mu\text{C}$) lies in the xy-plane. Calculate the potential at $z = 0.1 , \text{m}$.

• (a) $1.90 \times 10^5 , \text{V}$
• (b) $1.91 \times 10^5 , \text{V}$
• (c) $1.92 \times 10^5 , \text{V}$
• (d) $1.93 \times 10^5 , \text{V}$

98. A capacitor with $C = 15 , \mu\text{F}$ is charged to $V = 60 , \text{V}$. Calculate the energy stored.

• (a) $2.69 \times 10^{-2} , \text{J}$
• (b) $2.70 \times 10^{-2} , \text{J}$
• (c) $2.71 \times 10^{-2} , \text{J}$
• (d) $2.72 \times 10^{-2} , \text{J}$

99. A charge $q = 2 , \mu\text{C}$ moves between $V_a = 150 , \text{V}$ and $V_b = 50 , \text{V}$. Calculate the work done by the field.

• (a) $1.99 \times 10^{-4} , \text{J}$
• (b) $2.00 \times 10^{-4} , \text{J}$
• (c) $2.01 \times 10^{-4} , \text{J}$
• (d) $2.02 \times 10^{-4} , \text{J}$

100. A spherical conductor of radius $R = 0.1 , \text{m}$ has $Q = 4 , \mu\text{C}$. Calculate the potential on the surface.
     - (a) $3.59 \times 10^5 , \text{V}$
     - (b) $3.60 \times 10^5 , \text{V}$
     - (c) $3.61 \times 10^5 , \text{V}$
     - (d) $3.62 \times 10^5 , \text{V}$

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