Maxwell’s Equations; Magnetism of Matter Problems

This section provides 100 problems to test your understanding of Maxwell’s equations and the magnetism of matter, including calculations of electric and magnetic fields, induced emf, displacement current, and magnetic properties of materials. 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 point charge $Q = 5 , \mu\text{C}$ is at the center of a spherical Gaussian surface with radius $r = 0.2 , \text{m}$. Calculate the electric field $E$ at the surface ($\epsilon_0 = 8.85 \times 10^{-12} , \text{C}^2/(\text{N·m}^2)$).

   • (a) $1.12 \times 10^5 , \text{N/C}$
   • (b) $1.13 \times 10^5 , \text{N/C}$
   • (c) $1.14 \times 10^5 , \text{N/C}$
   • (d) $1.15 \times 10^5 , \text{N/C}$

2. An infinite sheet has a surface charge density $\sigma = 2 \times 10^{-6} , \text{C/m}^2$. Calculate the electric field $E$ near the sheet.

   • (a) $1.12 \times 10^5 , \text{N/C}$
   • (b) $1.13 \times 10^5 , \text{N/C}$
   • (c) $1.14 \times 10^5 , \text{N/C}$
   • (d) $1.15 \times 10^5 , \text{N/C}$

3. An infinite line charge has $\lambda = 3 \times 10^{-6} , \text{C/m}$. Calculate $E$ at a distance $r = 0.1 , \text{m}$.

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

4. A spherical shell with uniform charge density $\rho = 10^{-6} , \text{C/m}^3$ has radius $R = 0.3 , \text{m}$. Calculate $E$ at $r = 0.1 , \text{m}$ inside.

   • (a) $3.33 \times 10^2 , \text{N/C}$
   • (b) $3.34 \times 10^2 , \text{N/C}$
   • (c) $3.35 \times 10^2 , \text{N/C}$
   • (d) $3.36 \times 10^2 , \text{N/C}$

5. A bar magnet is enclosed in a spherical Gaussian surface. Calculate the net magnetic flux through the surface.

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

6. A loop with area $A = 0.05 , \text{m}^2$ is in a magnetic field $B = 0.6 t , \text{T}$ (increasing). Calculate the induced emf at $t = 1 , \text{s}$.

   • (a) $0.029 , \text{V}$
   • (b) $0.030 , \text{V}$
   • (c) $0.031 , \text{V}$
   • (d) $0.032 , \text{V}$

7. A solenoid with $N = 200$ turns, $A = 0.02 , \text{m}^2$, has $B$ increasing at $0.1 , \text{T/s}$. Calculate the induced emf.

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

8. A capacitor with plates of area $A = 0.01 , \text{m}^2$, separation $d = 0.001 , \text{m}$, has $E$ increasing at $10^6 , \text{V/m·s}$. Calculate the displacement current.

   • (a) $8.84 \times 10^{-8} , \text{A}$
   • (b) $8.85 \times 10^{-8} , \text{A}$
   • (c) $8.86 \times 10^{-8} , \text{A}$
   • (d) $8.87 \times 10^{-8} , \text{A}$

9. A material has magnetic susceptibility $\chi = -0.005$. Calculate the relative permeability $\mu_r$.

   • (a) $0.994$
   • (b) $0.995$
   • (c) $0.996$
   • (d) $0.997$

10. A paramagnetic material has $\chi = 0.01$. Calculate $\mu_r$.

    • (a) $1.009$
    • (b) $1.010$
    • (c) $1.011$
    • (d) $1.012$

11. A ferromagnetic material with $\mu_r = 1000$ is in a solenoid with $H = 50 , \text{A/m}$. Calculate $B$ ($\mu_0 = 4 \pi \times 10^{-7} , \text{H/m}$).

    • (a) $6.28 \times 10^{-2} , \text{T}$
    • (b) $6.29 \times 10^{-2} , \text{T}$
    • (c) $6.30 \times 10^{-2} , \text{T}$
    • (d) $6.31 \times 10^{-2} , \text{T}$

12. A hysteresis loop for a ferromagnetic material has an area of $0.02 , \text{J/m}^3$ per cycle, and the material volume is $0.1 , \text{m}^3$. Calculate the energy loss per cycle.

    • (a) $0.0019 , \text{J}$
    • (b) $0.0020 , \text{J}$
    • (c) $0.0021 , \text{J}$
    • (d) $0.0022 , \text{J}$

13. A point charge $Q = 8 , \mu\text{C}$ is at $r = 0.4 , \text{m}$. Calculate $E$.

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

14. An infinite sheet with $\sigma = 6 \times 10^{-6} , \text{C/m}^2$. Calculate $E$.

    • (a) $3.38 \times 10^5 , \text{N/C}$
    • (b) $3.39 \times 10^5 , \text{N/C}$
    • (c) $3.40 \times 10^5 , \text{N/C}$
    • (d) $3.41 \times 10^5 , \text{N/C}$

15. A line charge $\lambda = 1 \times 10^{-6} , \text{C/m}$ at $r = 0.02 , \text{m}$. Calculate $E$.

    • (a) $8.99 \times 10^5 , \text{N/C}$
    • (b) $9.00 \times 10^5 , \text{N/C}$
    • (c) $9.01 \times 10^5 , \text{N/C}$
    • (d) $9.02 \times 10^5 , \text{N/C}$

16. A spherical shell $\rho = 2 \times 10^{-6} , \text{C/m}^3$, $R = 0.5 , \text{m}$, at $r = 0.2 , \text{m}$ inside. Calculate $E$.

    • (a) $2.39 \times 10^2 , \text{N/C}$
    • (b) $2.40 \times 10^2 , \text{N/C}$
    • (c) $2.41 \times 10^2 , \text{N/C}$
    • (d) $2.42 \times 10^2 , \text{N/C}$

17. A closed surface encloses a solenoid with $B = 0.4 , \text{T}$ inside. Calculate the net magnetic flux.

    • (a) $0 , \text{Wb}$
    • (b) $1 \times 10^{-4} , \text{Wb}$
    • (c) $2 \times 10^{-4} , \text{Wb}$
    • (d) $3 \times 10^{-4} , \text{Wb}$

18. A loop $A = 0.03 , \text{m}^2$ in $B = 0.8 t , \text{T}$ (increasing). Calculate $\mathcal{E}$ at $t = 1 , \text{s}$.

    • (a) $0.023 , \text{V}$
    • (b) $0.024 , \text{V}$
    • (c) $0.025 , \text{V}$
    • (d) $0.026 , \text{V}$

19. A solenoid $N = 150$, $A = 0.015 , \text{m}^2$, $B$ increasing at $0.05 , \text{T/s}$. Calculate $\mathcal{E}$.

    • (a) $0.112 , \text{V}$
    • (b) $0.113 , \text{V}$
    • (c) $0.114 , \text{V}$
    • (d) $0.115 , \text{V}$

20. A capacitor $A = 0.02 , \text{m}^2$, $d = 0.002 , \text{m}$, $E$ increasing at $2 \times 10^6 , \text{V/m·s}$. Calculate $I_d$.

    • (a) $3.53 \times 10^{-7} , \text{A}$
    • (b) $3.54 \times 10^{-7} , \text{A}$
    • (c) $3.55 \times 10^{-7} , \text{A}$
    • (d) $3.56 \times 10^{-7} , \text{A}$

21. A material $\chi = -0.002$. Calculate $\mu_r$.

    • (a) $0.997$
    • (b) $0.998$
    • (c) $0.999$
    • (d) $1.000$

22. A material $\chi = 0.015$. Calculate $\mu_r$.

    • (a) $1.014$
    • (b) $1.015$
    • (c) $1.016$
    • (d) $1.017$

23. A material $\mu_r = 2000$, $H = 20 , \text{A/m}$. Calculate $B$.

    • (a) $5.02 \times 10^{-2} , \text{T}$
    • (b) $5.03 \times 10^{-2} , \text{T}$
    • (c) $5.04 \times 10^{-2} , \text{T}$
    • (d) $5.05 \times 10^{-2} , \text{T}$

24. A hysteresis loop area $0.05 , \text{J/m}^3$, volume $0.2 , \text{m}^3$. Calculate energy loss per cycle.

    • (a) $0.0099 , \text{J}$
    • (b) $0.0100 , \text{J}$
    • (c) $0.0101 , \text{J}$
    • (d) $0.0102 , \text{J}$

25. A point charge $Q = 1 , \mu\text{C}$ at $r = 0.05 , \text{m}$. Calculate $E$.

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

26. An infinite sheet $\sigma = 8 \times 10^{-6} , \text{C/m}^2$. Calculate $E$.

    • (a) $4.51 \times 10^5 , \text{N/C}$
    • (b) $4.52 \times 10^5 , \text{N/C}$
    • (c) $4.53 \times 10^5 , \text{N/C}$
    • (d) $4.54 \times 10^5 , \text{N/C}$

27. A line charge $\lambda = 4 \times 10^{-6} , \text{C/m}$ at $r = 0.08 , \text{m}$. Calculate $E$.

    • (a) $8.99 \times 10^5 , \text{N/C}$
    • (b) $9.00 \times 10^5 , \text{N/C}$
    • (c) $9.01 \times 10^5 , \text{N/C}$
    • (d) $9.02 \times 10^5 , \text{N/C}$

28. A spherical shell $\rho = 5 \times 10^{-6} , \text{C/m}^3$, $R = 0.4 , \text{m}$, at $r = 0.3 , \text{m}$ inside. Calculate $E$.

    • (a) $1.99 \times 10^3 , \text{N/C}$
    • (b) $2.00 \times 10^3 , \text{N/C}$
    • (c) $2.01 \times 10^3 , \text{N/C}$
    • (d) $2.02 \times 10^3 , \text{N/C}$

29. A loop $A = 0.04 , \text{m}^2$ in $B = 0.2 t , \text{T}$ (increasing). Calculate $\mathcal{E}$.

    • (a) $0.0079 , \text{V}$
    • (b) $0.0080 , \text{V}$
    • (c) $0.0081 , \text{V}$
    • (d) $0.0082 , \text{V}$

30. A solenoid $N = 300$, $A = 0.01 , \text{m}^2$, $B$ increasing at $0.03 , \text{T/s}$. Calculate $\mathcal{E}$.

    • (a) $0.0899 , \text{V}$
    • (b) $0.0900 , \text{V}$
    • (c) $0.0901 , \text{V}$
    • (d) $0.0902 , \text{V}$

31. A spacecraft capacitor $A = 0.015 , \text{m}^2$, $d = 0.0015 , \text{m}$, $E$ increasing at $1.5 \times 10^6 , \text{V/m·s}$. Calculate $I_d$ for shielding analysis.

    • (a) $1.99 \times 10^{-7} , \text{A}$
    • (b) $2.00 \times 10^{-7} , \text{A}$
    • (c) $2.01 \times 10^{-7} , \text{A}$
    • (d) $2.02 \times 10^{-7} , \text{A}$

32. A material $\chi = 0.008$. Calculate $\mu_r$.

    • (a) $1.007$
    • (b) $1.008$
    • (c) $1.009$
    • (d) $1.010$

33. A material $\mu_r = 1500$, $H = 30 , \text{A/m}$. Calculate $B$.

    • (a) $5.65 \times 10^{-2} , \text{T}$
    • (b) $5.66 \times 10^{-2} , \text{T}$
    • (c) $5.67 \times 10^{-2} , \text{T}$
    • (d) $5.68 \times 10^{-2} , \text{T}$

34. A hysteresis loop area $0.03 , \text{J/m}^3$, volume $0.15 , \text{m}^3$. Calculate energy loss per cycle.

    • (a) $0.0044 , \text{J}$
    • (b) $0.0045 , \text{J}$
    • (c) $0.0046 , \text{J}$
    • (d) $0.0047 , \text{J}$

35. A point charge $Q = 3 , \mu\text{C}$ at $r = 0.15 , \text{m}$. Calculate $E$.

    • (a) $1.19 \times 10^5 , \text{N/C}$
    • (b) $1.20 \times 10^5 , \text{N/C}$
    • (c) $1.21 \times 10^5 , \text{N/C}$
    • (d) $1.22 \times 10^5 , \text{N/C}$

Conceptual Problems

36. What does Gauss’s law for electricity relate?

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

37. What does Gauss’s law for magnetism imply?

• (a) Existence of magnetic monopoles
• (b) No magnetic monopoles exist
• (c) Electric field lines are closed
• (d) Magnetic field is zero everywhere

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

• (a) N·m²/C
• (b) T·m²
• (c) A/m
• (d) V/m

39. What happens to the magnetic flux through a closed surface if a magnetic monopole existed?

• (a) It would be zero
• (b) It would be non-zero
• (c) It would be infinite
• (d) It would depend on the surface shape

40. What does Faraday’s law describe?

• (a) Induced emf due to changing electric flux
• (b) Induced emf due to changing magnetic flux
• (c) Magnetic field due to enclosed charge
• (d) Electric field due to enclosed current

41. What does the Ampere-Maxwell law include that Ampere’s law does not?

• (a) Conduction current
• (b) Displacement current
• (c) Magnetic flux
• (d) Electric charge

42. What is the unit of magnetic susceptibility $\chi$?

• (a) Henry
• (b) Dimensionless
• (c) A/m
• (d) T

43. What happens to the magnetic field $B$ inside a diamagnetic material?

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

44. What does a large positive $\chi$ indicate about a material?

• (a) Diamagnetic
• (b) Paramagnetic
• (c) Ferromagnetic
• (d) Non-magnetic

45. What is the dimension of magnetic field intensity $H$?

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

46. What does a zero net magnetic flux through a closed surface indicate?

• (a) Presence of a magnetic monopole
• (b) Absence of magnetic monopoles
• (c) No magnetic field
• (d) No electric field

47. What is the significance of $\frac{Q_{\text{enc}}}{\epsilon_0}$?

• (a) Magnetic flux through a surface
• (b) Electric flux through a closed surface
• (c) Induced emf in a loop
• (d) Displacement current

48. What does the displacement current in the Ampere-Maxwell law account for?

• (a) Changing magnetic field
• (b) Changing electric field
• (c) Static electric field
• (d) Static magnetic field

49. What does a negative $\chi$ indicate about a material?

• (a) Paramagnetic
• (b) Ferromagnetic
• (c) Diamagnetic
• (d) Non-magnetic

50. How does ferromagnetism assist in spacecraft shielding?

• (a) Increases electric field
• (b) Enhances magnetic field shielding
• (c) Reduces magnetic susceptibility
• (d) Increases electric flux

Derivation Problems

51. Derive the electric field due to a point charge using Gauss’s law $E = \frac{Q}{4 \pi \epsilon_0 r^2}$.

52. Derive the electric field due to an infinite sheet $E = \frac{\sigma}{2 \epsilon_0}$.

53. Derive the electric field due to an infinite line charge $E = \frac{\lambda}{2 \pi \epsilon_0 r}$.

54. Derive the electric field inside a uniformly charged spherical shell using Gauss’s law.

55. Derive the magnetic flux through a closed surface using Gauss’s law for magnetism $\oint \vec{B} \cdot d\vec{A} = 0$.

56. Derive the induced emf in a loop due to a changing magnetic field using Faraday’s law $\mathcal{E} = -\frac{d\Phi_B}{dt}$.

57. Derive the displacement current in a charging capacitor using the Ampere-Maxwell law.

58. Derive the magnetic field $B$ in a material using $\vec{B} = \mu_0 (\vec{H} + \vec{M})$.

59. Derive the relationship between $\mu_r$ and $\chi$ for a material $\mu_r = 1 + \chi$.

60. Derive the electric field due to a uniformly charged sphere outside the sphere.

61. Derive the magnetic field inside a solenoid using the Ampere-Maxwell law.

62. Derive the energy loss per cycle in a hysteresis loop for a ferromagnetic material.

63. Derive the electric flux through a Gaussian surface enclosing multiple charges.

64. Derive the induced electric field around a loop using Faraday’s law $\oint \vec{E} \cdot d\vec{l} = -\frac{d\Phi_B}{dt}$.

65. Derive the magnetization $\vec{M}$ in a material using $\vec{M} = \chi \vec{H}$.

NEET-style Conceptual Problems

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

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

67. What does a non-zero electric flux through a closed surface indicate?

• (a) No charge enclosed
• (b) Net charge enclosed
• (c) No electric field
• (d) No magnetic field

68. What is the relationship between electric field and enclosed charge in Gauss’s law?

• (a) $E \propto \frac{1}{Q_{\text{enc}}}$
• (b) $E \propto Q_{\text{enc}}$
• (c) $E$ is independent of $Q_{\text{enc}}$
• (d) $E \propto Q_{\text{enc}}^2$

69. What happens to the magnetic flux through a closed surface if a magnetic dipole is enclosed?

• (a) Becomes non-zero
• (b) Remains zero
• (c) Becomes infinite
• (d) Depends on the dipole strength

70. What is the dimension of $\epsilon_0$?

• (a) $[\text{M}^{-1} \text{L}^{-3} \text{T}^4 \text{A}^2]$
• (b) $[\text{M} \text{L} \text{T}^{-1}]$
• (c) $[\text{L} \text{T}^{-2}]$
• (d) $[\text{M} \text{L}^2 \text{T}^{-1}]$

71. What does the induced emf in a loop depend on according to Faraday’s law?

• (a) Rate of change of electric flux
• (b) Rate of change of magnetic flux
• (c) Static magnetic field
• (d) Static electric field

72. What is the role of displacement current in the Ampere-Maxwell law?

• (a) Accounts for static electric fields
• (b) Accounts for changing electric fields
• (c) Reduces magnetic field
• (d) Increases conduction current

73. What happens to the magnetic field inside a paramagnetic material?

• (a) Decreases
• (b) Increases slightly
• (c) Becomes zero
• (d) Becomes infinite

74. Why does a ferromagnetic material exhibit hysteresis?
    .ConcurrentModificationException: (a) Due to alignment of magnetic domains

• (b) Due to static magnetic fields
• (c) Due to electric fields
• (d) Due to diamagnetism

75. What is the unit of magnetization $M$?

• (a) A/m
• (b) T
• (c) H/m
• (d) V/m

76. What does a constant electric field inside a conductor indicate?

• (a) Non-zero flux
• (b) Zero flux
• (c) No charges
• (d) Static charges

77. Which type of material has a large susceptibility $\chi$?

• (a) Diamagnetic
• (b) Paramagnetic
• (c) Ferromagnetic
• (d) Non-magnetic

78. What is the direction of the induced electric field in Faraday’s law?

• (a) Along the magnetic field
• (b) Perpendicular to the changing magnetic flux
• (c) Random
• (d) Zero

79. What does a pseudo-force do in a non-inertial frame for Maxwell’s equations?

• (a) Affects perceived fields
• (b) Affects charge distribution
• (c) Creates magnetic monopoles
• (d) Reduces susceptibility

80. 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) $[\text{M} \text{L}^2 \text{T}^{-1}]$

81. What is the role of ferromagnetic materials in spacecraft shielding?

• (a) Increase electric field
• (b) Enhance magnetic field shielding
• (c) Reduce voltage
• (d) Increase resistance

82. What happens to the electric field inside a conductor in electrostatic equilibrium?

• (a) Becomes non-zero
• (b) Becomes zero
• (c) Increases
• (d) Decreases

83. Why does the Ampere-Maxwell law include displacement current?

• (a) To account for static fields
• (b) To ensure continuity of magnetic fields
• (c) To reduce electric flux
• (d) To increase conduction current

84. What is the significance of $\mu_0 (1 + \chi)$?

• (a) Electric permittivity
• (b) Magnetic permeability of a material
• (c) Induced emf
• (d) Displacement current

85. What is the unit of displacement current?

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

86. 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 electric field

87. What is the physical significance of $\frac{\sigma}{\epsilon_0}$?

• (a) Electric field due to an infinite sheet
• (b) Magnetic field due to a current
• (c) Induced emf in a loop
• (d) Displacement current

88. Why does the magnetic field inside a diamagnetic material decrease?

• (a) Due to positive $\chi$
• (b) Due to negative $\chi$
• (c) Due to increased $H$
• (d) Due to decreased $H$

89. What is the dimension of $\frac{Q_{\text{enc}}}{\epsilon_0}$?

• (a) $[\text{M} \text{L}^3 \text{T}^{-2} \text{A}^{-1}]$
• (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 displacement current assist in spacecraft communication?

• (a) Increases voltage
• (b) Ensures field continuity in capacitors
• (c) Reduces current
• (d) Increases resistance

91. What is the role of enclosed charge in Gauss’s law for electricity?

• (a) $E \propto \frac{1}{Q_{\text{enc}}}$
• (b) $E \propto Q_{\text{enc}}$
• (c) No dependence
• (d) Exponential dependence

92. What does a high $\mu_r$ in a material indicate?

• (a) Diamagnetic
• (b) Paramagnetic
• (c) Ferromagnetic
• (d) Non-magnetic

93. What is the physical significance of $\mu_0 \epsilon_0 \frac{d\Phi_E}{dt}$?

• (a) Conduction current
• (b) Displacement current
• (c) Induced emf
• (d) Magnetic flux

94. What is the dimension of $\vec{B} \cdot d\vec{A}$?

• (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 a ferromagnetic material have a high susceptibility?

• (a) Due to alignment of magnetic domains
• (b) Due to static fields
• (c) Due to diamagnetic properties
• (d) Due to low $\mu_r$

NEET-style Numerical Problems

96. A point charge $Q = 4 , \mu\text{C}$ at $r = 0.25 , \text{m}$. Calculate $E$.

• (a) $5.75 \times 10^4 , \text{N/C}$
• (b) $5.76 \times 10^4 , \text{N/C}$
• (c) $5.77 \times 10^4 , \text{N/C}$
• (d) $5.78 \times 10^4 , \text{N/C}$

97. A loop $A = 0.02 , \text{m}^2$ in $B = 0.5 t , \text{T}$ (increasing). Calculate $\mathcal{E}$.

• (a) $0.0099 , \text{V}$
• (b) $0.0100 , \text{V}$
• (c) $0.0101 , \text{V}$
• (d) $0.0102 , \text{V}$

98. A solenoid $N = 100$, $A = 0.01 , \text{m}^2$, $B$ increasing at $0.04 , \text{T/s}$. Calculate $\mathcal{E}$.

• (a) $0.0399 , \text{V}$
• (b) $0.0400 , \text{V}$
• (c) $0.0401 , \text{V}$
• (d) $0.0402 , \text{V}$

99. A capacitor $A = 0.03 , \text{m}^2$, $d = 0.003 , \text{m}$, $E$ increasing at $10^6 , \text{V/m·s}$. Calculate $I_d$.

• (a) $2.65 \times 10^{-7} , \text{A}$
• (b) $2.66 \times 10^{-7} , \text{A}$
• (c) $2.67 \times 10^{-7} , \text{A}$
• (d) $2.68 \times 10^{-7} , \text{A}$

100. A material $\mu_r = 500$, $H = 40 , \text{A/m}$. Calculate $B$.
     - (a) $2.51 \times 10^{-2} , \text{T}$
     - (b) $2.52 \times 10^{-2} , \text{T}$
     - (c) $2.53 \times 10^{-2} , \text{T}$
     - (d) $2.54 \times 10^{-2} , \text{T}$

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