Induction and Inductance Problems

This section provides 100 problems to test your understanding of electromagnetic induction and inductance, including calculations of induced emf, inductance, energy storage, and circuit behavior in RL and LC circuits. 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 electromagnetism, a key topic for JEE/NEET success.

Numerical Problems

1. A loop with area $A=0.1{\text{m}}^2$ is in a magnetic field $B=0.4t\text{T}$ (increasing). Calculate the induced emf at $t=1\text{s}$.

   • (a) $0.039\text{V}$
   • (b) $0.040\text{V}$
   • (c) $0.041\text{V}$
   • (d) $0.042\text{V}$

2. A coil with $N=100$ turns, area $A=0.02{\text{m}}^2$, is in a field $B=0.5\text{T}$ decreasing at $0.1\text{T/s}$. Calculate the induced emf.

   • (a) $0.199\text{V}$
   • (b) $0.200\text{V}$
   • (c) $0.201\text{V}$
   • (d) $0.202\text{V}$

3. A rod of length $l=0.5\text{m}$ moves at $v=4\text{m/s}$ in a magnetic field $B=0.3\text{T}$ (perpendicular). Calculate the motional emf.

   • (a) $0.599\text{V}$
   • (b) $0.600\text{V}$
   • (c) $0.601\text{V}$
   • (d) $0.602\text{V}$

4. A solenoid with $N=200$ turns, length $l=0.4\text{m}$, area $A=0.01{\text{m}}^2$ (${\mu }_0=4\pi \times {10}^{-7}\text{T·m/A}$). Calculate the self-inductance $L$.

   • (a) $1.25\times {10}^{-3}\text{H}$
   • (b) $1.26\times {10}^{-3}\text{H}$
   • (c) $1.27\times {10}^{-3}\text{H}$
   • (d) $1.28\times {10}^{-3}\text{H}$

5. A coil with $L=0.2\text{H}$ has a current changing at $3\text{A/s}$. Calculate the induced emf.

   • (a) $0.599\text{V}$
   • (b) $0.600\text{V}$
   • (c) $0.601\text{V}$
   • (d) $0.602\text{V}$

6. Two coils have mutual inductance $M=0.03\text{H}$. The current in coil 1 changes at $2\text{A/s}$. Calculate the emf induced in coil 2.

   • (a) $0.059\text{V}$
   • (b) $0.060\text{V}$
   • (c) $0.061\text{V}$
   • (d) $0.062\text{V}$

7. An inductor $L=0.4\text{H}$ carries a current $I=2\text{A}$. Calculate the energy stored in the inductor.

   • (a) $0.799\text{J}$
   • (b) $0.800\text{J}$
   • (c) $0.801\text{J}$
   • (d) $0.802\text{J}$

8. An RL circuit has $L=0.1\text{H}$, $R=5\mathrm{\Omega }$, $\mathcal{E}=10\text{V}$. Calculate the current at $t=0.02\text{s}$ after the switch is closed.

   • (a) $0.632\text{A}$
   • (b) $0.633\text{A}$
   • (c) $0.634\text{A}$
   • (d) $0.635\text{A}$

9. An LC circuit has $L=0.2\text{H}$, $C=50\mu \text{F}$. Calculate the oscillation frequency $f$.

   • (a) $15.91\text{Hz}$
   • (b) $15.92\text{Hz}$
   • (c) $15.93\text{Hz}$
   • (d) $15.94\text{Hz}$

10. A loop with $A=0.05{\text{m}}^2$ rotates at $\omega =20\text{rad/s}$ in $B=0.1\text{T}$. Calculate the maximum induced emf.

    • (a) $0.049\text{V}$
    • (b) $0.050\text{V}$
    • (c) $0.051\text{V}$
    • (d) $0.052\text{V}$

11. A rod $l=0.6\text{m}$ moves at $v=3\text{m/s}$ in $B=0.2\text{T}$ at ${30}^{\circ }$. Calculate the motional emf.

    • (a) $0.179\text{V}$
    • (b) $0.180\text{V}$
    • (c) $0.181\text{V}$
    • (d) $0.182\text{V}$

12. A solenoid $N=150$, $l=0.3\text{m}$, $A=0.015{\text{m}}^2$. Calculate $L$.

    • (a) $1.12\times {10}^{-3}\text{H}$
    • (b) $1.13\times {10}^{-3}\text{H}$
    • (c) $1.14\times {10}^{-3}\text{H}$
    • (d) $1.15\times {10}^{-3}\text{H}$

13. A coil $L=0.3\text{H}$, current changes from $0$ to $4\text{A}$ in $0.2\text{s}$. Calculate $\mathcal{E}$.

    • (a) $5.99\text{V}$
    • (b) $6.00\text{V}$
    • (c) $6.01\text{V}$
    • (d) $6.02\text{V}$

14. Two coils $N_1=200$, $N_2=100$, $M=0.04\text{H}$, $I_1$ changes at $5\text{A/s}$. Calculate ${\mathcal{E}}_2$.

    • (a) $0.199\text{V}$
    • (b) $0.200\text{V}$
    • (c) $0.201\text{V}$
    • (d) $0.202\text{V}$

15. An inductor $L=0.5\text{H}$, $I=3\text{A}$. Calculate $U$.

    • (a) $2.249\text{J}$
    • (b) $2.250\text{J}$
    • (c) $2.251\text{J}$
    • (d) $2.252\text{J}$

16. An RL circuit $L=0.2\text{H}$, $R=8\mathrm{\Omega }$, $\mathcal{E}=16\text{V}$, at $t=0.025\text{s}$. Calculate $I$.

    • (a) $0.865\text{A}$
    • (b) $0.866\text{A}$
    • (c) $0.867\text{A}$
    • (d) $0.868\text{A}$

17. An LC circuit $L=0.1\text{H}$, $C=200\mu \text{F}$. Calculate $\omega$.

    • (a) $49.9\text{rad/s}$
    • (b) $50.0\text{rad/s}$
    • (c) $50.1\text{rad/s}$
    • (d) $50.2\text{rad/s}$

18. A loop $A=0.03{\text{m}}^2$ in $B=0.6t\text{T}$ (increasing). Calculate $\mathcal{E}$ at $t=1\text{s}$.

    • (a) $0.017\text{V}$
    • (b) $0.018\text{V}$
    • (c) $0.019\text{V}$
    • (d) $0.020\text{V}$

19. A coil $N=80$, $A=0.01{\text{m}}^2$, $B=0.2\text{T}$ decreasing at $0.05\text{T/s}$. Calculate $\mathcal{E}$.

    • (a) $0.039\text{V}$
    • (b) $0.040\text{V}$
    • (c) $0.041\text{V}$
    • (d) $0.042\text{V}$

20. A rod $l=0.4\text{m}$, $v=2\text{m/s}$, $B=0.5\text{T}$ (perpendicular). Calculate $\mathcal{E}$.

    • (a) $0.399\text{V}$
    • (b) $0.400\text{V}$
    • (c) $0.401\text{V}$
    • (d) $0.402\text{V}$

21. A solenoid $N=300$, $l=0.2\text{m}$, $A=0.005{\text{m}}^2$. Calculate $L$.

    • (a) $2.82\times {10}^{-3}\text{H}$
    • (b) $2.83\times {10}^{-3}\text{H}$
    • (c) $2.84\times {10}^{-3}\text{H}$
    • (d) $2.85\times {10}^{-3}\text{H}$

22. A coil $L=0.15\text{H}$, current changes from $2\text{A}$ to $5\text{A}$ in $0.3\text{s}$. Calculate $\mathcal{E}$.

    • (a) $1.49\text{V}$
    • (b) $1.50\text{V}$
    • (c) $1.51\text{V}$
    • (d) $1.52\text{V}$

23. Two coils $M=0.02\text{H}$, $I_1$ changes at $4\text{A/s}$. Calculate ${\mathcal{E}}_2$.

    • (a) $0.079\text{V}$
    • (b) $0.080\text{V}$
    • (c) $0.081\text{V}$
    • (d) $0.082\text{V}$

24. An inductor $L=0.6\text{H}$, $I=1.5\text{A}$. Calculate $U$.

    • (a) $0.674\text{J}$
    • (b) $0.675\text{J}$
    • (c) $0.676\text{J}$
    • (d) $0.677\text{J}$

25. An RL circuit $L=0.05\text{H}$, $R=10\mathrm{\Omega }$, $I_0=2\text{A}$, at $t=0.005\text{s}$ (decaying). Calculate $I$.

    • (a) $1.213\text{A}$
    • (b) $1.214\text{A}$
    • (c) $1.215\text{A}$
    • (d) $1.216\text{A}$

26. A loop $A=0.04{\text{m}}^2$ in $B=0.8\text{T}$ (constant) moves out at $v=3\text{m/s}$. Calculate $\mathcal{E}$.

    • (a) $0.095\text{V}$
    • (b) $0.096\text{V}$
    • (c) $0.097\text{V}$
    • (d) $0.098\text{V}$

27. A coil $N=120$, $A=0.015{\text{m}}^2$, $B=0.3\text{T}$ decreasing at $0.02\text{T/s}$. Calculate $\mathcal{E}$.

    • (a) $0.035\text{V}$
    • (b) $0.036\text{V}$
    • (c) $0.037\text{V}$
    • (d) $0.038\text{V}$

28. A rod $l=0.2\text{m}$, $v=5\text{m/s}$, $B=0.4\text{T}$ at ${60}^{\circ }$. Calculate $\mathcal{E}$.

    • (a) $0.345\text{V}$
    • (b) $0.346\text{V}$
    • (c) $0.347\text{V}$
    • (d) $0.348\text{V}$

29. A solenoid $N=250$, $l=0.5\text{m}$, $A=0.008{\text{m}}^2$. Calculate $L$.

    • (a) $1.00\times {10}^{-3}\text{H}$
    • (b) $1.01\times {10}^{-3}\text{H}$
    • (c) $1.02\times {10}^{-3}\text{H}$
    • (d) $1.03\times {10}^{-3}\text{H}$

30. A coil $L=0.25\text{H}$, current changes from $1\text{A}$ to $4\text{A}$ in $0.5\text{s}$. Calculate $\mathcal{E}$.

    • (a) $1.49\text{V}$
    • (b) $1.50\text{V}$
    • (c) $1.51\text{V}$
    • (d) $1.52\text{V}$

31. A spacecraft RL circuit has $L=0.1\text{H}$, $R=2\mathrm{\Omega }$, $\mathcal{E}=12\text{V}$, at $t=0.05\text{s}$. Calculate $I$ for power management.

    • (a) $3.78\text{A}$
    • (b) $3.79\text{A}$
    • (c) $3.80\text{A}$
    • (d) $3.81\text{A}$

32. An LC circuit $L=0.3\text{H}$, $C=300\mu \text{F}$. Calculate $f$.

    • (a) $10.58\text{Hz}$
    • (b) $10.59\text{Hz}$
    • (c) $10.60\text{Hz}$
    • (d) $10.61\text{Hz}$

33. A loop $A=0.06{\text{m}}^2$ rotates at $\omega =15\text{rad/s}$ in $B=0.2\text{T}$. Calculate the maximum $\mathcal{E}$.

    • (a) $0.179\text{V}$
    • (b) $0.180\text{V}$
    • (c) $0.181\text{V}$
    • (d) $0.182\text{V}$

34. An inductor $L=0.8\text{H}$, $I=1\text{A}$. Calculate $U$.

    • (a) $0.399\text{J}$
    • (b) $0.400\text{J}$
    • (c) $0.401\text{J}$
    • (d) $0.402\text{J}$

35. An RL circuit $L=0.4\text{H}$, $R=20\mathrm{\Omega }$, $I_0=1\text{A}$, at $t=0.02\text{s}$ (decaying). Calculate $I$.

    • (a) $0.367\text{A}$
    • (b) $0.368\text{A}$
    • (c) $0.369\text{A}$
    • (d) $0.370\text{A}$

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Conceptual Problems

36. What does Faraday’s law relate?

• (a) Induced emf to magnetic field strength
• (b) Induced emf to rate of change of magnetic flux
• (c) Induced current to electric field
• (d) Induced emf to static magnetic field

37. What does Lenz’s law determine?

• (a) Magnitude of induced emf
• (b) Direction of induced current
• (c) Rate of change of flux
• (d) Energy stored in the inductor

38. What is the unit of magnetic flux in SI units?

• (a) Tesla
• (b) Weber
• (c) Henry
• (d) Volt

39. What happens to the induced emf in a loop if the magnetic field increases?

• (a) Becomes zero
• (b) Induces a current to oppose the increase
• (c) Induces a current to support the increase
• (d) Remains constant

40. What does self-inductance measure?

• (a) Induced emf in another coil
• (b) Induced emf due to changing current in the same coil
• (c) Magnetic field strength
• (d) Electric field strength

41. What is the unit of inductance in SI units?

• (a) Henry
• (b) Weber
• (c) Tesla
• (d) Joule

42. What does mutual inductance describe?

• (a) Induced emf in the same coil
• (b) Induced emf in one coil due to current change in another
• (c) Magnetic flux in a single coil
• (d) Energy stored in an inductor

43. What happens to the current in an RL circuit immediately after closing the switch?

• (a) Reaches maximum instantly
• (b) Starts at zero and grows exponentially
• (c) Starts at maximum and decays
• (d) Oscillates

44. What does the time constant $\tau$ in an RL circuit represent?

• (a) Time to reach maximum current
• (b) Time to reach ~63% of maximum current
• (c) Time to fully discharge
• (d) Time to oscillate

45. What is the dimension of inductance $L$?

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

46. What does a zero induced emf in a loop indicate?

• (a) No magnetic field
• (b) No change in magnetic flux
• (c) Infinite flux
• (d) No current

47. What is the significance of $\frac{1}{2}L{I}^2$?

• (a) Induced emf in a coil
• (b) Energy stored in an inductor
• (c) Magnetic flux through a loop
• (d) Power in a circuit

48. What happens to the current in an LC circuit over time?

• (a) Grows exponentially
• (b) Decays exponentially
• (c) Oscillates
• (d) Remains constant

49. What does the frequency of an LC circuit depend on?

• (a) Resistance only
• (b) Inductance and capacitance
• (c) Current only
• (d) Voltage only

50. How do inductors function in spacecraft power systems?

• (a) Increase voltage
• (b) Store energy and regulate current changes
• (c) Reduce inductance
• (d) Increase resistance

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Derivation Problems

51. Derive Faraday’s law for a loop in a changing magnetic field $\mathcal{E}=-\frac{d{\mathrm{\Phi }}_B}{dt}$.

52. Derive the motional emf for a rod moving in a magnetic field $\mathcal{E}=Blv$.

53. Derive the self-inductance of a solenoid $L={\mu }_0\frac{N^2}{l}A$.

54. Derive the mutual inductance between two coils $M=\frac{N_2{\mathrm{\Phi }}_{B2}}{I_1}$.

55. Derive the energy stored in an inductor $U=\frac{1}{2}L{I}^2$.

56. Derive the current growth in an RL circuit $I=\frac{\mathcal{E}}{R}(1-e^{-t/\tau })$.

57. Derive the current decay in an RL circuit $I=I_0{e}^{-t/\tau }$.

58. Derive the oscillation frequency of an LC circuit $\omega =\frac{1}{\sqrt{LC}}$.

59. Derive the direction of induced current using Lenz’s law for a loop in an increasing magnetic field.

60. Derive the induced emf in a rotating loop in a uniform magnetic field $\mathcal{E}=BA\omega \sin (\omega t)$.

61. Derive the time constant of an RL circuit $\tau =\frac{L}{R}$.

62. Derive the flux linkage in a solenoid $N{\mathrm{\Phi }}_B={\mu }_0\frac{N^2}{l}IA$.

63. Derive the induced electric field from Faraday’s law $\oint \overrightarrow{E}\cdot d\overrightarrow{l}=-\frac{d{\mathrm{\Phi }}_B}{dt}$.

64. Derive the power dissipated in an RL circuit during current growth.

65. Derive the energy oscillation between $L$ and $C$ in an LC circuit.

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NEET-style Conceptual Problems

66. What is the unit of induced emf in SI units?

• (a) Volt
• (b) Ampere
• (c) Ohm
• (d) Henry

67. What does an increasing magnetic flux through a loop indicate?

• (a) Induced current opposes the increase
• (b) Induced current supports the increase
• (c) No induced current
• (d) Infinite current

68. What is the relationship between induced emf and rate of change of flux?

• (a) $\mathcal{E}\propto \frac{1}{\frac{d{\mathrm{\Phi }}_B}{dt}}$
• (b) $\mathcal{E}\propto \frac{d{\mathrm{\Phi }}_B}{dt}$
• (c) $\mathcal{E}$ is independent of $\frac{d{\mathrm{\Phi }}_B}{dt}$
• (d) $\mathcal{E}\propto {\left(\frac{d{\mathrm{\Phi }}_B}{dt}\right)}^2$

69. What happens to the induced current in a loop if the magnetic field decreases?

• (a) Produces a field to oppose the decrease
• (b) Produces a field to support the decrease
• (c) Becomes zero
• (d) Becomes infinite

70. What is the dimension of mutual inductance $M$?

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

71. What does the self-inductance of a coil depend on?

• (a) Current through the coil
• (b) Geometry and number of turns
• (c) Resistance of the coil
• (d) Voltage across the coil

72. What is the role of Lenz’s law in electromagnetic induction?

• (a) Increases the induced emf
• (b) Determines the direction of induced current
• (c) Reduces the magnetic field
• (d) Increases the resistance

73. What happens to the current in an RL circuit after a long time?

• (a) Becomes zero
• (b) Reaches a steady value
• (c) Oscillates
• (d) Becomes infinite

74. Why does the current in an LC circuit oscillate?

• (a) Due to energy exchange between $L$ and $C$
• (b) Due to resistance
• (c) Due to constant voltage
• (d) Due to static magnetic field

75. What is the unit of the time constant $\tau$ in an RL circuit?

• (a) Second
• (b) Ohm
• (c) Henry
• (d) Volt

76. What does a constant current in an inductor indicate?

• (a) Induced emf is zero
• (b) Induced emf is maximum
• (c) Magnetic field is zero
• (d) Energy is zero

77. Which type of circuit exhibits oscillatory behavior?

• (a) RL circuit
• (b) RC circuit
• (c) LC circuit
• (d) Pure resistive circuit

78. What is the direction of the induced current in a loop moving into a magnetic field?

• (a) Produces a field in the same direction as $B$
• (b) Produces a field opposite to $B$
• (c) No current
• (d) Random direction

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

• (a) Affects perceived emf
• (b) Affects charge distribution
• (c) Creates magnetic field
• (d) Reduces inductance

80. What is the dimension of ${\mathrm{\Phi }}_B$?

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

81. What is the role of inductors in spacecraft power systems?

• (a) Increase voltage
• (b) Regulate current changes and store energy
• (c) Reduce inductance
• (d) Increase resistance

82. What happens to the energy in an LC circuit over time?

• (a) Dissipates completely
• (b) Oscillates between $L$ and $C$
• (c) Remains constant in $L$
• (d) Remains constant in $C$

83. Why does the induced emf in a coil depend on the rate of change of current?

• (a) Due to $\mathcal{E}=-L\frac{dI}{dt}$
• (b) Due to increased resistance
• (c) Due to decreased capacitance
• (d) Due to static field

84. What is the significance of $Blv$?

• (a) Induced emf in a rotating loop
• (b) Motional emf in a moving conductor
• (c) Energy in an inductor
• (d) Magnetic flux in a coil

85. What is the unit of energy stored in an inductor?

• (a) Joule
• (b) Volt
• (c) Ampere
• (d) Weber

86. What does a zero current in an RL circuit after a long time (decaying) indicate?

• (a) Circuit is still active
• (b) Inductor has fully discharged
• (c) Resistance is zero
• (d) Inductance is infinite

87. What is the physical significance of ${\mu }_0\frac{N^2}{l}A$?

• (a) Induced emf in a coil
• (b) Self-inductance of a solenoid
• (c) Mutual inductance
• (d) Energy in an inductor

88. Why does the current in an RL circuit grow exponentially?

• (a) Due to $I=\frac{\mathcal{E}}{R}(1-e^{-t/\tau })$
• (b) Due to oscillations
• (c) Due to constant voltage
• (d) Due to static field

89. What is the dimension of $\frac{1}{\sqrt{LC}}$?

• (a) $[{\text{T}}^{-1}]$
• (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 mutual inductance assist in spacecraft communication?

• (a) Increases current
• (b) Couples signals between coils
• (c) Reduces voltage
• (d) Increases resistance

91. What is the role of the number of turns in a solenoid’s inductance?

• (a) $L\propto N$
• (b) $L\propto N^2$
• (c) No dependence
• (d) $L\propto \frac{1}{N}$

92. What does a high induced emf in a loop indicate?

• (a) Low rate of flux change
• (b) High rate of flux change
• (c) No flux change
• (d) Constant flux

93. What is the physical significance of $\frac{L}{R}$?

• (a) Frequency of an LC circuit
• (b) Time constant of an RL circuit
• (c) Energy in an inductor
• (d) Induced emf

94. What is the dimension of $BA$?

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

95. Why does the energy in an inductor depend on $I^2$?

• (a) Due to $U=\frac{1}{2}L{I}^2$
• (b) Due to symmetry
• (c) Due to field lines
• (d) Due to charge quantization

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NEET-style Numerical Problems

96. A loop $A=0.02{\text{m}}^2$ in $B=0.3t\text{T}$ (increasing). Calculate $\mathcal{E}$ at $t=1\text{s}$.

• (a) $0.005\text{V}$
• (b) $0.006\text{V}$
• (c) $0.007\text{V}$
• (d) $0.008\text{V}$

97. A rod $l=0.3\text{m}$, $v=2\text{m/s}$, $B=0.4\text{T}$ (perpendicular). Calculate $\mathcal{E}$.

• (a) $0.239\text{V}$
• (b) $0.240\text{V}$
• (c) $0.241\text{V}$
• (d) $0.242\text{V}$

98. A solenoid $N=100$, $l=0.2\text{m}$, $A=0.01{\text{m}}^2$. Calculate $L$.

• (a) $6.28\times {10}^{-4}\text{H}$
• (b) $6.29\times {10}^{-4}\text{H}$
• (c) $6.30\times {10}^{-4}\text{H}$
• (d) $6.31\times {10}^{-4}\text{H}$

99. An RL circuit $L=0.1\text{H}$, $R=4\mathrm{\Omega }$, $\mathcal{E}=8\text{V}$, at $t=0.025\text{s}$. Calculate $I$.

• (a) $0.865\text{A}$
• (b) $0.866\text{A}$
• (c) $0.867\text{A}$
• (d) $0.868\text{A}$

100. An LC circuit $L=0.5\text{H}$, $C=500\mu \text{F}$. Calculate $f$.
     - (a) $10.06\text{Hz}$
     - (b) $10.07\text{Hz}$
     - (c) $10.08\text{Hz}$
     - (d) $10.09\text{Hz}$

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