Magnetic Fields Due to Currents Problems

This section provides 100 problems to test your understanding of magnetic fields generated by currents, including calculations using the Biot-Savart law and Ampere’s law, fields due to specific current configurations, superposition, and their applications. 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 current element $I = 5 , \text{A}$, $dl = 0.02 , \text{m}$ is at a distance $r = 0.1 , \text{m}$ (perpendicular). Calculate the magnetic field $dB$ ($\mu_0 = 4 \pi \times 10^{-7} , \text{T·m/A}$).

   • (a) $9.99 \times 10^{-9} , \text{T}$
   • (b) $1.00 \times 10^{-8} , \text{T}$
   • (c) $1.01 \times 10^{-8} , \text{T}$
   • (d) $1.02 \times 10^{-8} , \text{T}$

2. An infinite straight wire carries $I = 8 , \text{A}$. Calculate the magnetic field at $r = 0.04 , \text{m}$ using the Biot-Savart law.

   • (a) $3.99 \times 10^{-5} , \text{T}$
   • (b) $4.00 \times 10^{-5} , \text{T}$
   • (c) $4.01 \times 10^{-5} , \text{T}$
   • (d) $4.02 \times 10^{-5} , \text{T}$

3. A circular loop of radius $R = 0.05 , \text{m}$ carries $I = 3 , \text{A}$. Calculate the magnetic field at the center of the loop.

   • (a) $3.76 \times 10^{-5} , \text{T}$
   • (b) $3.77 \times 10^{-5} , \text{T}$
   • (c) $3.78 \times 10^{-5} , \text{T}$
   • (d) $3.79 \times 10^{-5} , \text{T}$

4. A solenoid with $n = 1000 , \text{turns/m}$ carries $I = 0.2 , \text{A}$. Calculate the magnetic field inside using Ampere’s law.

   • (a) $2.51 \times 10^{-4} , \text{T}$
   • (b) $2.52 \times 10^{-4} , \text{T}$
   • (c) $2.53 \times 10^{-4} , \text{T}$
   • (d) $2.54 \times 10^{-4} , \text{T}$

5. A toroid with $N = 800$ turns, mean radius $r = 0.4 , \text{m}$, carries $I = 0.5 , \text{A}$. Calculate the magnetic field inside the toroid.

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

6. A finite straight wire extends from $x = -0.1 , \text{m}$ to $x = 0.1 , \text{m}$ with $I = 4 , \text{A}$, observed at $y = 0.05 , \text{m}$. Calculate the magnetic field.

   • (a) $1.13 \times 10^{-5} , \text{T}$
   • (b) $1.14 \times 10^{-5} , \text{T}$
   • (c) $1.15 \times 10^{-5} , \text{T}$
   • (d) $1.16 \times 10^{-5} , \text{T}$

7. A circular loop with $R = 0.03 , \text{m}$, $I = 2 , \text{A}$ is observed at $x = 0.04 , \text{m}$ on its axis. Calculate the magnetic field.

   • (a) $2.87 \times 10^{-6} , \text{T}$
   • (b) $2.88 \times 10^{-6} , \text{T}$
   • (c) $2.89 \times 10^{-6} , \text{T}$
   • (d) $2.90 \times 10^{-6} , \text{T}$

8. An infinite current sheet has surface current density $K = 1500 , \text{A/m}$. Calculate the magnetic field on one side.

   • (a) $9.42 \times 10^{-4} , \text{T}$
   • (b) $9.43 \times 10^{-4} , \text{T}$
   • (c) $9.44 \times 10^{-4} , \text{T}$
   • (d) $9.45 \times 10^{-4} , \text{T}$

9. Two parallel wires, each with $I = 5 , \text{A}$ (same direction), are separated by $d = 0.1 , \text{m}$. Calculate the magnetic field at the midpoint between them.

   • (a) $0 , \text{T}$
   • (b) $1.00 \times 10^{-5} , \text{T}$
   • (c) $2.00 \times 10^{-5} , \text{T}$
   • (d) $4.00 \times 10^{-5} , \text{T}$

10. Two parallel wires with $I_1 = 3 , \text{A}$, $I_2 = 3 , \text{A}$ (opposite directions) are separated by $d = 0.06 , \text{m}$. Calculate the magnetic field at the midpoint.

    • (a) $1.99 \times 10^{-5} , \text{T}$
    • (b) $2.00 \times 10^{-5} , \text{T}$
    • (c) $2.01 \times 10^{-5} , \text{T}$
    • (d) $2.02 \times 10^{-5} , \text{T}$

11. A current element $I = 6 , \text{A}$, $dl = 0.01 , \text{m}$, at $r = 0.2 , \text{m}$ (perpendicular). Calculate $dB$.

    • (a) $2.99 \times 10^{-9} , \text{T}$
    • (b) $3.00 \times 10^{-9} , \text{T}$
    • (c) $3.01 \times 10^{-9} , \text{T}$
    • (d) $3.02 \times 10^{-9} , \text{T}$

12. An infinite wire with $I = 12 , \text{A}$ at $r = 0.03 , \text{m}$. Calculate $B$ using Ampere’s law.

    • (a) $7.99 \times 10^{-5} , \text{T}$
    • (b) $8.00 \times 10^{-5} , \text{T}$
    • (c) $8.01 \times 10^{-5} , \text{T}$
    • (d) $8.02 \times 10^{-5} , \text{T}$

13. A solenoid with $n = 500 , \text{turns/m}$, $I = 0.6 , \text{A}$. Calculate $B$ inside.

    • (a) $3.76 \times 10^{-4} , \text{T}$
    • (b) $3.77 \times 10^{-4} , \text{T}$
    • (c) $3.78 \times 10^{-4} , \text{T}$
    • (d) $3.79 \times 10^{-4} , \text{T}$

14. A toroid with $N = 1200$ turns, $r = 0.3 , \text{m}$, $I = 0.4 , \text{A}$. Calculate $B$ inside.

    • (a) $3.19 \times 10^{-4} , \text{T}$
    • (b) $3.20 \times 10^{-4} , \text{T}$
    • (c) $3.21 \times 10^{-4} , \text{T}$
    • (d) $3.22 \times 10^{-4} , \text{T}$

15. A finite wire from $x = -0.15 , \text{m}$ to $x = 0.15 , \text{m}$, $I = 2 , \text{A}$, at $y = 0.1 , \text{m}$. Calculate $B$.

    • (a) $5.65 \times 10^{-6} , \text{T}$
    • (b) $5.66 \times 10^{-6} , \text{T}$
    • (c) $5.67 \times 10^{-6} , \text{T}$
    • (d) $5.68 \times 10^{-6} , \text{T}$

16. A circular loop with $R = 0.02 , \text{m}$, $I = 5 , \text{A}$ at its center. Calculate $B$.

    • (a) $1.570 \times 10^{-4} , \text{T}$
    • (b) $1.571 \times 10^{-4} , \text{T}$
    • (c) $1.572 \times 10^{-4} , \text{T}$
    • (d) $1.573 \times 10^{-4} , \text{T}$

17. A current sheet with $K = 2000 , \text{A/m}$. Calculate $B$ on one side.

    • (a) $1.256 \times 10^{-3} , \text{T}$
    • (b) $1.257 \times 10^{-3} , \text{T}$
    • (c) $1.258 \times 10^{-3} , \text{T}$
    • (d) $1.259 \times 10^{-3} , \text{T}$

18. Two parallel wires with $I_1 = 4 , \text{A}$, $I_2 = 2 , \text{A}$ (same direction), $d = 0.08 , \text{m}$, at $x = 0.04 , \text{m}$. Calculate $B$.

    • (a) $0 , \text{T}$
    • (b) $1.00 \times 10^{-5} , \text{T}$
    • (c) $2.00 \times 10^{-5} , \text{T}$
    • (d) $3.00 \times 10^{-5} , \text{T}$

19. A current element $I = 7 , \text{A}$, $dl = 0.03 , \text{m}$, at $r = 0.15 , \text{m}$ (perpendicular). Calculate $dB$.

    • (a) $9.32 \times 10^{-9} , \text{T}$
    • (b) $9.33 \times 10^{-9} , \text{T}$
    • (c) $9.34 \times 10^{-9} , \text{T}$
    • (d) $9.35 \times 10^{-9} , \text{T}$

20. An infinite wire with $I = 10 , \text{A}$ at $r = 0.05 , \text{m}$. Calculate $B$.

    • (a) $3.99 \times 10^{-5} , \text{T}$
    • (b) $4.00 \times 10^{-5} , \text{T}$
    • (c) $4.01 \times 10^{-5} , \text{T}$
    • (d) $4.02 \times 10^{-5} , \text{T}$

21. A circular loop with $R = 0.06 , \text{m}$, $I = 1 , \text{A}$, at $x = 0.08 , \text{m}$. Calculate $B$.

    • (a) $7.53 \times 10^{-7} , \text{T}$
    • (b) $7.54 \times 10^{-7} , \text{T}$
    • (c) $7.55 \times 10^{-7} , \text{T}$
    • (d) $7.56 \times 10^{-7} , \text{T}$

22. A solenoid with $n = 600 , \text{turns/m}$, $I = 0.8 , \text{A}$. Calculate $B$ inside.

    • (a) $6.03 \times 10^{-4} , \text{T}$
    • (b) $6.04 \times 10^{-4} , \text{T}$
    • (c) $6.05 \times 10^{-4} , \text{T}$
    • (d) $6.06 \times 10^{-4} , \text{T}$

23. A toroid with $N = 500$ turns, $r = 0.2 , \text{m}$, $I = 0.6 , \text{A}$. Calculate $B$ inside.

    • (a) $2.99 \times 10^{-4} , \text{T}$
    • (b) $3.00 \times 10^{-4} , \text{T}$
    • (c) $3.01 \times 10^{-4} , \text{T}$
    • (d) $3.02 \times 10^{-4} , \text{T}$

24. A finite wire from $x = -0.2 , \text{m}$ to $x = 0.2 , \text{m}$, $I = 3 , \text{A}$, at $y = 0.2 , \text{m}$. Calculate $B$.

    • (a) $2.12 \times 10^{-6} , \text{T}$
    • (b) $2.13 \times 10^{-6} , \text{T}$
    • (c) $2.14 \times 10^{-6} , \text{T}$
    • (d) $2.15 \times 10^{-6} , \text{T}$

25. Two parallel wires with $I_1 = 6 , \text{A}$, $I_2 = 6 , \text{A}$ (opposite directions), $d = 0.1 , \text{m}$, at $x = 0.05 , \text{m}$. Calculate $B$.

    • (a) $4.79 \times 10^{-5} , \text{T}$
    • (b) $4.80 \times 10^{-5} , \text{T}$
    • (c) $4.81 \times 10^{-5} , \text{T}$
    • (d) $4.82 \times 10^{-5} , \text{T}$

26. A current element $I = 2 , \text{A}$, $dl = 0.04 , \text{m}$, at $r = 0.05 , \text{m}$ (perpendicular). Calculate $dB$.

    • (a) $3.19 \times 10^{-8} , \text{T}$
    • (b) $3.20 \times 10^{-8} , \text{T}$
    • (c) $3.21 \times 10^{-8} , \text{T}$
    • (d) $3.22 \times 10^{-8} , \text{T}$

27. An infinite wire with $I = 15 , \text{A}$ at $r = 0.02 , \text{m}$. Calculate $B$.

    • (a) $1.499 \times 10^{-4} , \text{T}$
    • (b) $1.500 \times 10^{-4} , \text{T}$
    • (c) $1.501 \times 10^{-4} , \text{T}$
    • (d) $1.502 \times 10^{-4} , \text{T}$

28. A circular loop with $R = 0.04 , \text{m}$, $I = 3 , \text{A}$ at its center. Calculate $B$.

    • (a) $4.71 \times 10^{-5} , \text{T}$
    • (b) $4.72 \times 10^{-5} , \text{T}$
    • (c) $4.73 \times 10^{-5} , \text{T}$
    • (d) $4.74 \times 10^{-5} , \text{T}$

29. A solenoid with $n = 400 , \text{turns/m}$, $I = 1.2 , \text{A}$. Calculate $B$ inside.

    • (a) $6.03 \times 10^{-4} , \text{T}$
    • (b) $6.04 \times 10^{-4} , \text{T}$
    • (c) $6.05 \times 10^{-4} , \text{T}$
    • (d) $6.06 \times 10^{-4} , \text{T}$

30. A toroid with $N = 600$ turns, $r = 0.15 , \text{m}$, $I = 0.5 , \text{A}$. Calculate $B$ inside.

    • (a) $3.99 \times 10^{-4} , \text{T}$
    • (b) $4.00 \times 10^{-4} , \text{T}$
    • (c) $4.01 \times 10^{-4} , \text{T}$
    • (d) $4.02 \times 10^{-4} , \text{T}$

31. A spacecraft wire with $I = 2 , \text{A}$ and a loop $R = 0.05 , \text{m}$, $I = 1 , \text{A}$ at $x = 0.1 , \text{m}$, find $B$ at $x = 0.05 , \text{m}$.

    • (a) $5.72 \times 10^{-6} , \text{T}$
    • (b) $5.73 \times 10^{-6} , \text{T}$
    • (c) $5.74 \times 10^{-6} , \text{T}$
    • (d) $5.75 \times 10^{-6} , \text{T}$

32. A finite wire from $x = -0.05 , \text{m}$ to $x = 0.05 , \text{m}$, $I = 6 , \text{A}$, at $y = 0.03 , \text{m}$. Calculate $B$.

    • (a) $2.82 \times 10^{-5} , \text{T}$
    • (b) $2.83 \times 10^{-5} , \text{T}$
    • (c) $2.84 \times 10^{-5} , \text{T}$
    • (d) $2.85 \times 10^{-5} , \text{T}$

33. A circular loop with $R = 0.01 , \text{m}$, $I = 10 , \text{A}$ at its center. Calculate $B$.

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

34. A solenoid with $n = 200 , \text{turns/m}$, $I = 2 , \text{A}$. Calculate $B$ inside.

    • (a) $5.02 \times 10^{-4} , \text{T}$
    • (b) $5.03 \times 10^{-4} , \text{T}$
    • (c) $5.04 \times 10^{-4} , \text{T}$
    • (d) $5.05 \times 10^{-4} , \text{T}$

35. A toroid with $N = 1000$ turns, $r = 0.5 , \text{m}$, $I = 0.2 , \text{A}$. Calculate $B$ inside.

    • (a) $7.99 \times 10^{-5} , \text{T}$
    • (b) $8.00 \times 10^{-5} , \text{T}$
    • (c) $8.01 \times 10^{-5} , \text{T}$
    • (d) $8.02 \times 10^{-5} , \text{T}$

Conceptual Problems

36. What does the Biot-Savart law calculate?

• (a) Electric field due to a current
• (b) Magnetic field due to a current element
• (c) Electric field due to a charge
• (d) Magnetic field due to a static charge

37. What does Ampere’s law relate?

• (a) Magnetic field to enclosed charge
• (b) Magnetic field to enclosed current
• (c) Electric field to enclosed current
• (d) Electric field to enclosed charge

38. What is the relationship between the magnetic field and distance from an infinite straight wire?

• (a) $B \propto r$
• (b) $B \propto \frac{1}{r}$
• (c) $B \propto r^2$
• (d) $B \propto \frac{1}{r^2}$

39. What happens to the magnetic field at the center of a circular loop as the radius increases?

• (a) Increases
• (b) Decreases
• (c) Remains constant
• (d) Becomes zero

40. What is the magnetic field inside an ideal solenoid?

• (a) Zero
• (b) Uniform and non-zero
• (c) Proportional to radius
• (d) Proportional to distance from the axis

41. What is the unit of magnetic field in SI units?

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

42. What does the right-hand rule determine in the context of the Biot-Savart law?

• (a) Direction of electric field
• (b) Direction of magnetic field due to a current
• (c) Direction of current flow
• (d) Direction of charge motion

43. What happens to the magnetic field outside an ideal solenoid?

• (a) Uniform and non-zero
• (b) Proportional to distance
• (c) Approximately zero
• (d) Infinite

44. What does the magnetic field inside a toroid depend on?

• (a) Distance from the center
• (b) Total current only
• (c) Radius of the toroid only
• (d) No dependence

45. What is the dimension of magnetic field $B$?

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

46. What does a zero magnetic field at a point indicate?

• (a) No current or magnetic sources nearby
• (b) Infinite current
• (c) Infinite magnetic field elsewhere
• (d) No electric field

47. What is the significance of $\frac{\mu_0 I}{2 \pi r}$?

• (a) Magnetic field due to a circular loop
• (b) Magnetic field due to an infinite straight wire
• (c) Magnetic field inside a solenoid
• (d) Magnetic field inside a toroid

48. What happens to the magnetic field on the axis of a circular loop as you move away from the center?

• (a) Increases
• (b) Decreases
• (c) Remains constant
• (d) Becomes zero immediately

49. What does the uniform magnetic field on either side of a current sheet indicate?

• (a) Field depends on distance
• (b) Field is independent of distance
• (c) Field is zero
• (d) Field is infinite

50. How do magnetic fields due to currents assist in spacecraft navigation systems?

• (a) Increase current flow
• (b) Detect orientation using magnetic sensors
• (c) Reduce resistance
• (d) Increase voltage

Derivation Problems

51. Derive the magnetic field due to a small current element using the Biot-Savart law $dB = \frac{\mu_0 I dl \sin \theta}{4 \pi r^2}$.

52. Derive the magnetic field due to an infinite straight wire using the Biot-Savart law $B = \frac{\mu_0 I}{2 \pi r}$.

53. Derive the magnetic field at the center of a circular loop $B = \frac{\mu_0 I}{2 R}$.

54. Derive the magnetic field due to an infinite straight wire using Ampere’s law $B = \frac{\mu_0 I}{2 \pi r}$.

55. Derive the magnetic field inside a solenoid using Ampere’s law $B = \mu_0 n I$.

56. Derive the magnetic field inside a toroid $B = \frac{\mu_0 N I}{2 \pi r}$.

57. Derive the magnetic field due to a finite straight wire $B = \frac{\mu_0 I}{4 \pi a} (\sin \theta_2 - \sin \theta_1)$.

58. Derive the magnetic field on the axis of a circular loop $B = \frac{\mu_0 I R^2}{2 (R^2 + x^2)^{3/2}}$.

59. Derive the magnetic field due to an infinite current sheet $B = \frac{\mu_0 K}{2}$.

60. Derive the net magnetic field between two parallel infinite wires using superposition.

61. Derive the Biot-Savart law expression $d\vec{B} = \frac{\mu_0}{4 \pi} \frac{I (d\vec{l} \times \hat{r})}{r^2}$.

62. Derive the direction of the magnetic field using the right-hand rule for a current element.

63. Derive the magnetic field due to a semi-infinite straight wire using the Biot-Savart law.

64. Derive the magnetic field at a point due to two perpendicular current elements using superposition.

65. Derive the cancellation of magnetic fields at the midpoint between two parallel wires with currents in the same direction.

NEET-style Conceptual Problems

66. What is the unit of permeability of free space $\mu_0$?

• (a) $\text{T·m/A}$
• (b) $\text{N/A}$
• (c) $\text{V·m/A}$
• (d) $\text{A/m}$

67. What does a circular magnetic field line around a wire indicate?

• (a) Magnetic field due to a static charge
• (b) Magnetic field due to a current
• (c) Electric field due to a current
• (d) Electric field due to a charge

68. What is the relationship between magnetic field and current in a solenoid?

• (a) $B \propto \frac{1}{I}$
• (b) $B \propto I$
• (c) $B$ is independent of $I$
• (d) $B \propto I^2$

69. What happens to the magnetic field inside a toroid as the radius increases?

• (a) Increases
• (b) Decreases
• (c) Remains constant
• (d) Becomes zero

70. What is the dimension of $\mu_0$?

• (a) $[\text{M} \text{L} \text{T}^{-2} \text{A}^{-2}]$
• (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 magnetic field at the center of a circular loop depend on?

• (a) Radius only
• (b) Current and radius
• (c) Distance from the center
• (d) No dependence

72. What is the role of symmetry in Ampere’s law?

• (a) Increases the magnetic field
• (b) Simplifies the calculation of the magnetic field
• (c) Reduces the current
• (d) Increases the resistance

73. What happens to the magnetic field outside a toroid?

• (a) Uniform and non-zero
• (b) Proportional to distance
• (c) Approximately zero
• (d) Infinite

74. Why does the magnetic field at the center of a circular loop depend on the radius?

• (a) Due to $B \propto \frac{1}{R}$
• (b) Due to increased current
• (c) Due to decreased current
• (d) Due to symmetry

75. What is the unit of surface current density $K$?

• (a) $\text{A/m}$
• (b) $\text{A/m}^2$
• (c) $\text{T}$
• (d) $\text{V}$

76. What does a constant magnetic field inside a solenoid indicate?

• (a) Non-uniform current
• (b) Uniform field along the axis
• (c) Zero field outside
• (d) Infinite field

77. Which type of current distribution produces a uniform magnetic field inside?

• (a) Straight wire
• (b) Circular loop
• (c) Solenoid
• (d) Current sheet

78. What is the direction of the magnetic field due to a current-carrying wire?

• (a) Along the wire
• (b) Perpendicular to the wire, circular around it
• (c) Random
• (d) Zero

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

• (a) Affects perceived field
• (b) Affects current distribution
• (c) Creates magnetic field
• (d) Reduces field

80. What is the dimension of $I d\vec{l} \times \hat{r}$ in the Biot-Savart law?

• (a) $[\text{A} \text{L}]$
• (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 magnetic fields due to currents in spacecraft navigation?

• (a) Increase current
• (b) Detect orientation using sensors
• (c) Reduce voltage
• (d) Increase resistance

82. What happens to the magnetic field between two parallel wires with currents in opposite directions?

• (a) Cancels out
• (b) Adds constructively
• (c) Becomes zero everywhere
• (d) Becomes infinite

83. Why does the magnetic field due to a current sheet not depend on distance?

• (a) Due to infinite extent and symmetry
• (b) Due to increased current
• (c) Due to decreased current
• (d) Due to finite size

84. What is the significance of $\frac{\mu_0 I R^2}{2 (R^2 + x^2)^{3/2}}$?

• (a) Magnetic field due to a straight wire
• (b) Magnetic field on the axis of a circular loop
• (c) Magnetic field inside a solenoid
• (d) Magnetic field inside a toroid

85. What is the unit of magnetic field due to a current?

• (a) Tesla
• (b) Volt
• (c) Ampere
• (d) Watt

86. What does a zero magnetic field at the center of a loop indicate?

• (a) No current
• (b) Infinite current
• (c) Infinite radius
• (d) No magnetic field outside

87. What is the physical significance of $\mu_0 n I$?

• (a) Magnetic field due to a straight wire
• (b) Magnetic field inside a solenoid
• (c) Magnetic field on the axis of a loop
• (d) Magnetic field due to a current sheet

88. Why does the magnetic field outside a solenoid approach zero for an ideal solenoid?

• (a) Due to cancellation of fields
• (b) Due to increased current
• (c) Due to decreased current
• (d) Due to symmetry

89. What is the dimension of $\frac{\mu_0 I}{r}$?

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

90. How does the magnetic field due to currents assist in spacecraft shielding?

• (a) Increases current
• (b) Deflects charged particles using magnetic fields
• (c) Reduces voltage
• (d) Increases resistance

91. What is the role of distance in the magnetic field due to a straight wire?

• (a) $B \propto r$
• (b) $B \propto \frac{1}{r}$
• (c) No dependence
• (d) Exponential dependence

92. What does a high magnetic field at the center of a loop indicate?

• (a) Large radius
• (b) Small radius or high current
• (c) No current
• (d) Infinite field

93. What is the physical significance of $\frac{\mu_0 K}{2}$?

• (a) Magnetic field due to a straight wire
• (b) Magnetic field due to a current sheet
• (c) Magnetic field inside a solenoid
• (d) Magnetic field on the axis of a loop

94. What is the dimension of $\vec{B} \cdot d\vec{l}$ in Ampere’s law?

• (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 magnetic field inside a toroid depend on the radius?

• (a) Due to $B \propto \frac{1}{r}$
• (b) Due to symmetry
• (c) Due to field lines
• (d) Due to current quantization

NEET-style Numerical Problems

96. A current element $I = 4 , \text{A}$, $dl = 0.02 , \text{m}$, at $r = 0.1 , \text{m}$ (perpendicular). Calculate $dB$.

• (a) $7.99 \times 10^{-9} , \text{T}$
• (b) $8.00 \times 10^{-9} , \text{T}$
• (c) $8.01 \times 10^{-9} , \text{T}$
• (d) $8.02 \times 10^{-9} , \text{T}$

97. An infinite wire with $I = 5 , \text{A}$ at $r = 0.02 , \text{m}$. Calculate $B$.

• (a) $4.99 \times 10^{-5} , \text{T}$
• (b) $5.00 \times 10^{-5} , \text{T}$
• (c) $5.01 \times 10^{-5} , \text{T}$
• (d) $5.02 \times 10^{-5} , \text{T}$

98. A circular loop with $R = 0.03 , \text{m}$, $I = 2 , \text{A}$ at its center. Calculate $B$.

• (a) $4.18 \times 10^{-5} , \text{T}$
• (b) $4.19 \times 10^{-5} , \text{T}$
• (c) $4.20 \times 10^{-5} , \text{T}$
• (d) $4.21 \times 10^{-5} , \text{T}$

99. A solenoid with $n = 300 , \text{turns/m}$, $I = 1.5 , \text{A}$. Calculate $B$ inside.

• (a) $5.65 \times 10^{-4} , \text{T}$
• (b) $5.66 \times 10^{-4} , \text{T}$
• (c) $5.67 \times 10^{-4} , \text{T}$
• (d) $5.68 \times 10^{-4} , \text{T}$

100. Two parallel wires with $I_1 = 2 , \text{A}$, $I_2 = 2 , \text{A}$ (opposite directions), $d = 0.04 , \text{m}$, at $x = 0.02 , \text{m}$. Calculate $B$.
     - (a) $1.99 \times 10^{-5} , \text{T}$
     - (b) $2.00 \times 10^{-5} , \text{T}$
     - (c) $2.01 \times 10^{-5} , \text{T}$
     - (d) $2.02 \times 10^{-5} , \text{T}$

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