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 $z=0.1\text{m}$ (${ϵ}_0=8.85\times {10}^{-12}{\text{C}}^2/{\text{N·m}}^2$).

   • (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 $V=\frac{Qd}{{ϵ}_{0}A}$.

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.

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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 $\frac{Qd}{{ϵ}_{0}A}$?

• (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 $\overrightarrow{E}=-\mathrm{\nabla }V$
• (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 $-\mathrm{\nabla }V$?

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

94. What is the dimension of $\overrightarrow{E}\cdot \overrightarrow{l}$ 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

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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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