Conduction of Electricity in Solids Problems

This section provides 100 problems to test your understanding of conduction in solids, including calculations of band gaps, carrier concentrations, conductivity, diode currents, and transistor gains, as well as applications like semiconductor devices in spacecraft electronics. 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 solid-state physics, a key topic for JEE/NEET success.

Numerical Problems

1. A conductor has an electron density $n = 8 \times 10^{28} , \text{m}^{-3}$ and relaxation time $\tau = 2 \times 10^{-14} , \text{s}$. Calculate its conductivity in S/m.

   • (a) $4.61 \times 10^7$
   • (b) $4.62 \times 10^7$
   • (c) $4.63 \times 10^7$
   • (d) $4.64 \times 10^7$

2. An intrinsic semiconductor has $E_g = 1.2 , \text{eV}$ at $T = 300 , \text{K}$. Calculate the exponential factor $e^{-E_g / 2 k T}$ ($k = 8.617 \times 10^{-5} , \text{eV/K}$).

   • (a) $e^{-23.2}$
   • (b) $e^{-23.3}$
   • (c) $e^{-23.4}$
   • (d) $e^{-23.5}$

3. A p-n junction diode has $I_S = 10^{-12} , \text{A}$, $V = 0.6 , \text{V}$ at $T = 300 , \text{K}$. Calculate the current $I$ in A ($k = 8.617 \times 10^{-5} , \text{eV/K}$).

   • (a) $1.02 \times 10^{-2}$
   • (b) $1.03 \times 10^{-2}$
   • (c) $1.04 \times 10^{-2}$
   • (d) $1.05 \times 10^{-2}$

4. An npn transistor has a current gain $\beta = 100$ and base current $I_B = 15 , \mu\text{A}$. Calculate the collector current $I_C$ in mA.

   • (a) 1.49 mA
   • (b) 1.50 mA
   • (c) 1.51 mA
   • (d) 1.52 mA

5. An n-type semiconductor has a donor concentration $N_D = 10^{16} , \text{cm}^{-3}$. Calculate the electron concentration in m$^{-3}$.

   • (a) $10^{22}$
   • (b) $10^{23}$
   • (c) $10^{24}$
   • (d) $10^{25}$

6. A silicon semiconductor ($E_g = 1.1 , \text{eV}$) at $T = 400 , \text{K}$. Calculate $E_g / 2 k T$ ($k = 8.617 \times 10^{-5} , \text{eV/K}$).

   • (a) 15.9
   • (b) 16.0
   • (c) 16.1
   • (d) 16.2

7. A p-type semiconductor has an acceptor concentration $N_A = 5 \times 10^{15} , \text{cm}^{-3}$. Calculate the hole concentration in m$^{-3}$.

   • (a) $5 \times 10^{21}$
   • (b) $5 \times 10^{22}$
   • (c) $5 \times 10^{23}$
   • (d) $5 \times 10^{24}$

8. A diode in forward bias has $I_S = 10^{-10} , \text{A}$, $V = 0.5 , \text{V}$ at $T = 300 , \text{K}$. Calculate $I$ in A.

   • (a) $1.79 \times 10^{-2}$
   • (b) $1.80 \times 10^{-2}$
   • (c) $1.81 \times 10^{-2}$
   • (d) $1.82 \times 10^{-2}$

9. A conductor has $\sigma = 6 \times 10^7 , \text{S/m}$, $n = 9 \times 10^{28} , \text{m}^{-3}$. Calculate $\tau$ in seconds.

   • (a) $2.59 \times 10^{-14}$
   • (b) $2.60 \times 10^{-14}$
   • (c) $2.61 \times 10^{-14}$
   • (d) $2.62 \times 10^{-14}$

10. An intrinsic semiconductor has $E_g = 0.7 , \text{eV}$ at $T = 300 , \text{K}$. Calculate $e^{-E_g / 2 k T}$.

    • (a) $e^{-13.5}$
    • (b) $e^{-13.6}$
    • (c) $e^{-13.7}$
    • (d) $e^{-13.8}$

11. A pnp transistor has $\beta = 80$, $I_B = 25 , \mu\text{A}$. Calculate $I_C$ in mA.

    • (a) 1.99 mA
    • (b) 2.00 mA
    • (c) 2.01 mA
    • (d) $2.02$ mA

12. An n-type semiconductor has $N_D = 2 \times 10^{14} , \text{cm}^{-3}$. Calculate $n_e$ in m$^{-3}$.

    • (a) $2 \times 10^{20}$
    • (b) $2 \times 10^{21}$
    • (c) $2 \times 10^{22}$
    • (d) $2 \times 10^{23}$

13. A diode in reverse bias has $I_S = 10^{-11} , \text{A}$. Calculate the current $I$ in A.

    • (a) $-10^{-11}$
    • (b) $-10^{-12}$
    • (c) $-10^{-13}$
    • (d) $-10^{-14}$

14. A germanium semiconductor ($E_g = 0.67 , \text{eV}$) at $T = 350 , \text{K}$. Calculate $E_g / 2 k T$.

    • (a) 11.1
    • (b) 11.2
    • (c) 11.3
    • (d) 11.4

15. A conductor has $n = 6 \times 10^{28} , \text{m}^{-3}$, $\tau = 3 \times 10^{-14} , \text{s}$. Calculate $\sigma$ in S/m.

    • (a) $5.18 \times 10^7$
    • (b) $5.19 \times 10^7$
    • (c) $5.20 \times 10^7$
    • (d) $5.21 \times 10^7$

16. A p-type semiconductor has $N_A = 10^{17} , \text{cm}^{-3}$. Calculate $n_h$ in m$^{-3}$.

    • (a) $10^{23}$
    • (b) $10^{24}$
    • (c) $10^{25}$
    • (d) $10^{26}$

17. A diode has $I_S = 10^{-12} , \text{A}$, $V = 0.8 , \text{V}$ at $T = 300 , \text{K}$. Calculate $I$ in A.

    • (a) $1.99 \times 10^{-1}$
    • (b) $2.00 \times 10^{-1}$
    • (c) $2.01 \times 10^{-1}$
    • (d) $2.02 \times 10^{-1}$

18. An npn transistor has $\beta = 200$, $I_B = 10 , \mu\text{A}$. Calculate $I_C$ in mA.

    • (a) 1.99 mA
    • (b) 2.00 mA
    • (c) 2.01 mA
    • (d) 2.02 mA

19. An intrinsic semiconductor has $E_g = 1.5 , \text{eV}$ at $T = 500 , \text{K}$. Calculate $e^{-E_g / 2 k T}$.

    • (a) $e^{-8.7}$
    • (b) $e^{-8.8}$
    • (c) $e^{-8.9}$
    • (d) $e^{-9.0}$

20. A conductor has $\sigma = 5 \times 10^7 , \text{S/m}$, $n = 7 \times 10^{28} , \text{m}^{-3}$. Calculate $\tau$ in seconds.

    • (a) $2.77 \times 10^{-14}$
    • (b) $2.78 \times 10^{-14}$
    • (c) $2.79 \times 10^{-14}$
    • (d) $2.80 \times 10^{-14}$

21. A p-type semiconductor has $N_A = 10^{16} , \text{cm}^{-3}$. Calculate $n_h$ in m$^{-3}$.

    • (a) $10^{22}$
    • (b) $10^{23}$
    • (c) $10^{24}$
    • (d) $10^{25}$

22. A diode has $I_S = 10^{-10} , \text{A}$, $V = 0.4 , \text{V}$ at $T = 300 , \text{K}$. Calculate $I$ in A.

    • (a) $2.84 \times 10^{-3}$
    • (b) $2.85 \times 10^{-3}$
    • (c) $2.86 \times 10^{-3}$
    • (d) $2.87 \times 10^{-3}$

23. An npn transistor has $\beta = 150$, $I_B = 30 , \mu\text{A}$. Calculate $I_C$ in mA.

    • (a) 4.49 mA
    • (b) 4.50 mA
    • (c) 4.51 mA
    • (d) 4.52 mA

24. An n-type semiconductor has $N_D = 10^{15} , \text{cm}^{-3}$. Calculate $n_e$ in m$^{-3}$.

    • (a) $10^{21}$
    • (b) $10^{22}$
    • (c) $10^{23}$
    • (d) $10^{24}$

25. A silicon semiconductor ($E_g = 1.1 , \text{eV}$) at $T = 250 , \text{K}$. Calculate $E_g / 2 k T$.

    • (a) 25.5
    • (b) 25.6
    • (c) 25.7
    • (d) 25.8

26. A conductor has $n = 5 \times 10^{28} , \text{m}^{-3}$, $\tau = 4 \times 10^{-14} , \text{s}$. Calculate $\sigma$ in S/m.

    • (a) $5.75 \times 10^7$
    • (b) $5.76 \times 10^7$
    • (c) $5.77 \times 10^7$
    • (d) $5.78 \times 10^7$

27. A p-type semiconductor has $N_A = 10^{14} , \text{cm}^{-3}$. Calculate $n_h$ in m$^{-3}$.

    • (a) $10^{20}$
    • (b) $10^{21}$
    • (c) $10^{22}$
    • (d) $10^{23}$

28. A diode has $I_S = 10^{-11} , \text{A}$, $V = 0.3 , \text{V}$ at $T = 300 , \text{K}$. Calculate $I$ in A.

    • (a) $1.16 \times 10^{-4}$
    • (b) $1.17 \times 10^{-4}$
    • (c) $1.18 \times 10^{-4}$
    • (d) $1.19 \times 10^{-4}$

29. An npn transistor has $\beta = 120$, $I_B = 40 , \mu\text{A}$. Calculate $I_C$ in mA.

    • (a) 4.79 mA
    • (b) 4.80 mA
    • (c) 4.81 mA
    • (d) 4.82 mA

30. An intrinsic semiconductor has $E_g = 1.0 , \text{eV}$ at $T = 200 , \text{K}$. Calculate $e^{-E_g / 2 k T}$.

    • (a) $e^{-29.0}$
    • (b) $e^{-29.1}$
    • (c) $e^{-29.2}$
    • (d) $e^{-29.3}$

31. A spacecraft semiconductor device uses n-type silicon with $N_D = 10^{16} , \text{cm}^{-3}$. Calculate $n_e$ in m$^{-3}$.

    • (a) $10^{22}$
    • (b) $10^{23}$
    • (c) $10^{24}$
    • (d) $10^{25}$

32. A conductor has $\sigma = 4 \times 10^7 , \text{S/m}$, $n = 8 \times 10^{28} , \text{m}^{-3}$. Calculate $\tau$ in seconds.

    • (a) $1.94 \times 10^{-14}$
    • (b) $1.95 \times 10^{-14}$
    • (c) $1.96 \times 10^{-14}$
    • (d) $1.97 \times 10^{-14}$

33. A p-type semiconductor has $N_A = 10^{18} , \text{cm}^{-3}$. Calculate $n_h$ in m$^{-3}$.

    • (a) $10^{24}$
    • (b) $10^{25}$
    • (c) $10^{26}$
    • (d) $10^{27}$

34. A diode has $I_S = 10^{-12} , \text{A}$, $V = 0.7 , \text{V}$ at $T = 300 , \text{K}$. Calculate $I$ in A.

    • (a) $5.79 \times 10^{-2}$
    • (b) $5.80 \times 10^{-2}$
    • (c) $5.81 \times 10^{-2}$
    • (d) $5.82 \times 10^{-2}$

35. An npn transistor has $\beta = 90$, $I_B = 50 , \mu\text{A}$. Calculate $I_C$ in mA.

    • (a) 4.49 mA
    • (b) 4.50 mA
    • (c) 4.51 mA
    • (d) 4.52 mA

Conceptual Problems

36. What does the band gap $E_g$ represent in solids?

    • (a) Energy difference between valence and conduction bands
    • (b) Fermi energy level
    • (c) Electron density
    • (d) Conductivity

37. What classifies a material as a conductor?

    • (a) Large band gap
    • (b) Overlapping valence and conduction bands
    • (c) Moderate band gap
    • (d) No Fermi energy

38. What is the unit of conductivity $\sigma$ in SI units?

    • (a) S/m
    • (b) Ohm
    • (c) Joule
    • (d) Watt

39. What happens to the conductivity of a semiconductor as temperature increases?

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

40. What are the majority carriers in an n-type semiconductor?

    • (a) Holes
    • (b) Electrons
    • (c) Both electrons and holes
    • (d) Neither electrons nor holes

41. What is the unit of carrier concentration $n$ in SI units?

    • (a) m$^{-3}$
    • (b) Joule
    • (c) Hertz
    • (d) Watt

42. What does a large band gap $E_g$ indicate about a material?

    • (a) Conductor
    • (b) Semiconductor
    • (c) Insulator
    • (d) Superconductor

43. What happens to the conductivity of a conductor as temperature increases?

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

44. What does doping do to a semiconductor?

    • (a) Increases the band gap
    • (b) Increases conductivity by adding charge carriers
    • (c) Decreases conductivity
    • (d) Removes charge carriers

45. What is the dimension of $\sigma$ in the Drude model?

    • (a) $[\text{M}^{-1} \text{L}^{-3} \text{T}^3 \text{I}^2]$
    • (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 forward bias do in a p-n junction diode?

    • (a) Increases the potential barrier
    • (b) Reduces the potential barrier
    • (c) Stops current flow
    • (d) Creates minority carriers

47. What is the significance of $I_S$ in the diode equation?

    • (a) Forward current
    • (b) Saturation current
    • (c) Reverse current
    • (d) Breakdown current

48. What happens to the current in a diode under reverse bias?

    • (a) Increases exponentially
    • (b) Becomes nearly constant and small
    • (c) Becomes zero
    • (d) Becomes infinite

49. What does a transistor do in amplification mode?

    • (a) Stops current flow
    • (b) Small base current controls large collector current
    • (c) Acts as a switch
    • (d) Reduces current

50. How are semiconductors used in spacecraft electronics?

    • (a) Increase resistance
    • (b) Enable sensors and processors through diodes and transistors
    • (c) Reduce conductivity
    • (d) Increase band gap

Derivation Problems

51. Derive the conductivity $\sigma = \frac{n e^2 \tau}{m}$ for a conductor using the Drude model.

52. Derive the carrier concentration relation $n_e = n_h \propto e^{-E_g / 2 k T}$ for an intrinsic semiconductor.

53. Derive the ideal diode equation $I = I_S \left(e^{e V / k T} - 1\right)$.

54. Derive the amplification relation $I_C = \beta I_B$ for a transistor.

55. Derive the band gap effect on carrier concentration in a semiconductor.

56. Derive the conductivity $\sigma$ for a conductor with given $n$ and $\tau$.

57. Derive the electron concentration $n_e \approx N_D$ in an n-type semiconductor.

58. Derive the current $I$ in a diode for a given forward bias voltage $V$.

59. Derive the collector current $I_C$ for a transistor with given $\beta$ and $I_B$.

60. Derive the hole concentration $n_h \approx N_A$ in a p-type semiconductor.

61. Derive the exponential factor $e^{-E_g / 2 k T}$ for an intrinsic semiconductor.

62. Derive the conductivity $\sigma$ for a conductor with given $\sigma$ and $n$.

63. Derive the current $I$ in a diode under reverse bias.

64. Derive the carrier concentration relation $n_e n_h = n_i^2$ for a semiconductor.

65. Derive the temperature dependence of conductivity in a semiconductor.

NEET-style Conceptual Problems

66. What is the unit of the band gap $E_g$ in SI units?

    • (a) eV
    • (b) Radian
    • (c) Hertz
    • (d) Watt

67. What does a zero band gap $E_g$ indicate about a material?

    • (a) Insulator
    • (b) Semiconductor
    • (c) Conductor
    • (d) Superconductor

68. What is the relationship between conductivity $\sigma$ and temperature $T$ in a semiconductor?

    • (a) $\sigma \propto \frac{1}{T}$
    • (b) $\sigma$ increases with $T$
    • (c) $\sigma$ is independent of $T$
    • (d) $\sigma \propto T$

69. What happens to the number of charge carriers in an intrinsic semiconductor as $T$ increases?

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

70. What is the dimension of $n e^2 \tau / m$?

    • (a) $[\text{M}^{-1} \text{L}^{-3} \text{T}^3 \text{I}^2]$
    • (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 Fermi energy $E_F$ determine in a solid?

    • (a) Band gap
    • (b) Highest occupied energy level at $T = 0 , \text{K}$
    • (c) Conductivity
    • (d) Electron density

72. What is the role of diodes in spacecraft electronics?

    • (a) Increase resistance
    • (b) Enable rectification and signal processing
    • (c) Reduce conductivity
    • (d) Increase band gap

73. What happens to the conductivity of an insulator as $T$ increases?

    • (a) Increases significantly
    • (b) Increases slightly
    • (c) Remains the same
    • (d) Becomes zero

74. Why does doping increase the conductivity of a semiconductor?

    • (a) Increases the band gap
    • (b) Adds extra charge carriers
    • (c) Reduces charge carriers
    • (d) Decreases temperature

75. What is the unit of saturation current $I_S$ in a diode?

    • (a) Ampere
    • (b) Joule
    • (c) Hertz
    • (d) Watt

76. What does a large $e V / k T$ in the diode equation indicate?

    • (a) Small current
    • (b) Large forward current
    • (c) No current
    • (d) Constant current

77. Which carriers are minority in a p-type semiconductor?

    • (a) Electrons
    • (b) Holes
    • (c) Both electrons and holes
    • (d) Neither electrons nor holes

78. What is the effect of reverse bias on a p-n junction diode?

    • (a) Increases current
    • (b) Reduces current to a small leakage
    • (c) Stops current completely
    • (d) Creates forward bias

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

    • (a) Affects perceived carrier concentration
    • (b) Affects band gap
    • (c) Creates diodes
    • (d) Reduces conductivity

80. What is the dimension of $e^{-E_g / 2 k T}$?

    • (a) Dimensionless
    • (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 transistors in spacecraft control systems?

    • (a) Increase resistance
    • (b) Amplify signals and act as switches
    • (c) Reduce conductivity
    • (d) Increase band gap

82. What happens to the current in a diode under forward bias?

    • (a) Increases exponentially
    • (b) Decreases exponentially
    • (c) Remains the same
    • (d) Becomes zero

83. Why are semiconductors used in electronics?

    • (a) Due to large band gap
    • (b) Due to controllable conductivity via doping
    • (c) Due to no band gap
    • (d) Due to high resistance

84. What is the significance of $\beta$ in a transistor?

    • (a) Saturation current
    • (b) Current gain
    • (c) Band gap
    • (d) Conductivity

85. What is the unit of $\tau$ in the Drude model?

    • (a) Second
    • (b) Joule
    • (c) Hertz
    • (d) Watt

86. What does a high conductivity $\sigma$ indicate about a material?

    • (a) Insulator
    • (b) Semiconductor
    • (c) Conductor
    • (d) Superconductor

87. What is the physical significance of $e^{e V / k T}$?

    • (a) Band gap factor
    • (b) Exponential increase in diode current
    • (c) Conductivity factor
    • (d) Carrier concentration

88. Why do insulators have negligible conductivity?

    • (a) Due to small band gap
    • (b) Due to large band gap
    • (c) Due to overlapping bands
    • (d) Due to doping

89. What is the dimension of $I_S (e^{e V / k T} - 1)$?

    • (a) $[\text{I}]$
    • (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 band theory explain conduction in solids?

    • (a) Through overlapping orbitals
    • (b) Through energy bands and gaps
    • (c) Through doping
    • (d) Through transistors

91. What is the role of the valence band in a solid?

    • (a) Contains conduction electrons
    • (b) Contains valence electrons
    • (c) Determines band gap
    • (d) Determines Fermi energy

92. What does a small $I_S$ in a diode indicate?

    • (a) Large forward current
    • (b) Small leakage current in reverse bias
    • (c) No current
    • (d) Constant current

93. What is the physical significance of $n_i^2$?

    • (a) Conductivity
    • (b) Product of electron and hole concentrations
    • (c) Band gap
    • (d) Fermi energy

94. What is the dimension of $k T$?

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

95. Why are transistors used in amplifiers?

    • (a) To reduce current
    • (b) To control large currents with small inputs
    • (c) To increase band gap
    • (d) To decrease conductivity

NEET-style Numerical Problems

96. An intrinsic semiconductor has $E_g = 1.1 , \text{eV}$ at $T = 300 , \text{K}$. Calculate $e^{-E_g / 2 k T}$.

    • (a) $e^{-21.2}$
    • (b) $e^{-21.3}$
    • (c) $e^{-21.4}$
    • (d) $e^{-21.5}$

97. A p-n junction diode has $I_S = 10^{-12} , \text{A}$, $V = 0.5 , \text{V}$ at $T = 300 , \text{K}$. Calculate $I$ in A.

    • (a) $1.02 \times 10^{-3}$
    • (b) $1.03 \times 10^{-3}$
    • (c) $1.04 \times 10^{-3}$
    • (d) $1.05 \times 10^{-3}$

98. An npn transistor has $\beta = 50$, $I_B = 20 , \mu\text{A}$. Calculate $I_C$ in mA.

    • (a) 0.99 mA
    • (b) 1.00 mA
    • (c) 1.01 mA
    • (d) 1.02 mA

99. An n-type semiconductor has $N_D = 10^{17} , \text{cm}^{-3}$. Calculate $n_e$ in m$^{-3}$.

    • (a) $10^{23}$
    • (b) $10^{24}$
    • (c) $10^{25}$
    • (d) $10^{26}$

100. A conductor has $n = 7 \times 10^{28} , \text{m}^{-3}$, $\tau = 2 \times 10^{-14} , \text{s}$. Calculate $\sigma$ in S/m.
     - (a) $4.02 \times 10^7$
     - (b) $4.03 \times 10^7$
     - (c) $4.04 \times 10^7$
     - (d) $4.05 \times 10^7$

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