Questions & explanations
1. A charging capacitor has conduction current I_c in the wire. Which statement about the displacement current I_d between the plates is correct?
- I_d = I_c and it produces a magnetic field in the opposite direction to I_c
- I_d = I_c and it produces a magnetic field in the same direction as I_c
- I_d = 0 and no magnetic field is produced between the plates
- I_d = I_c but it does not produce any magnetic field
Answer: I_d = I_c and it produces a magnetic field in the same direction as I_c
During charging, the displacement current I_d = ε₀ dΦ_E/dt equals the conduction current I_c. The Ampere-Maxwell law states that both I_c and I_d contribute to the magnetic field. For a parallel-plate capacitor, the magnetic field due to I_d is in the same direction as that due to I_c, ensuring continuity.
2. A parallel-plate capacitor is being charged by a steady current I. For a circular loop around the wire, Ampere's law gives different values of ∮B·dl for two different surfaces bounded by the loop. This is because:
- the enclosed current is different for the two surfaces
- the magnetic field is not constant on the loop
- the loop is not perpendicular to the wire
- the electric field between the plates is zero
Answer: the enclosed current is different for the two surfaces
Ampere's law ∮B·dl = μ₀ I_enc depends on the current passing through the chosen surface. For a flat surface cutting the wire, I_enc = I. For a curved surface passing between the plates, I_enc = 0. Thus the same loop gives two different values, exposing the incompleteness of classical Ampere's law.
3. In an electromagnetic wave in vacuum, the average energy density of the electric field is U_E and that of the magnetic field is U_B. Which relation holds?
- U_E = (1/2) U_B
- U_E = 2 U_B
- U_E = U_B
- U_E = c U_B
Answer: U_E = U_B
For an EM wave in vacuum, the electric and magnetic field amplitudes are related by E₀ = c B₀. The average energy densities are (1/2)ε₀ E_rms² and (1/2μ₀) B_rms². Using E_rms = c B_rms and c = 1/√(μ₀ε₀), both expressions simplify to (1/2)ε₀ E_rms², so U_E = U_B.
4. The Ampere-Maxwell law in integral form is:
- ∮ B·dl = μ₀ I_c
- ∮ B·dl = μ₀ ε₀ dΦ_E/dt
- ∮ B·dl = μ₀ (I_c + ε₀ dΦ_E/dt)
- ∮ B·dl = μ₀ (I_c - ε₀ dΦ_E/dt)
Answer: ∮ B·dl = μ₀ (I_c + ε₀ dΦ_E/dt)
The Ampere-Maxwell law states that the line integral of magnetic field around a closed loop equals μ₀ times the sum of conduction current and displacement current. Displacement current is ε₀ dΦ_E/dt, so the correct equation is ∮ B·dl = μ₀ (I_c + ε₀ dΦ_E/dt).
5. Sunlight (visible and UV) warms Earth's surface, which then emits infrared radiation. Why does this lead to atmospheric heating?
- Infrared radiation from the Sun is absorbed directly by greenhouse gases.
- Greenhouse gases reflect all incoming sunlight back into space.
- The Earth's surface emits only visible light, which is absorbed by the atmosphere.
- Greenhouse gases absorb the outgoing infrared and re-radiate some back to the surface.
Answer: Greenhouse gases absorb the outgoing infrared and re-radiate some back to the surface.
The greenhouse effect occurs because Earth's surface absorbs visible/UV sunlight and re-emits infrared radiation. Greenhouse gases like CO₂ and H₂O absorb this outgoing infrared and re-radiate some back to the surface, causing additional warming.
6. A plane electromagnetic wave traveling along the +x-axis has its electric field along which direction?
- Along the x-axis
- Along the y-axis
- Along the z-axis
- Along the direction of propagation
Answer: Along the y-axis
In a plane electromagnetic wave, the electric field (E), magnetic field (B), and propagation direction are mutually perpendicular. By convention, for propagation along +x, E is taken along y-axis and B along z-axis. Thus, E is along y-axis.
7. The electric field in a plane EM wave is E = E₀ sin(kx - ωt). What is the time-averaged total energy density ⟨u⟩?
- (1/4)ε₀E₀²
- ε₀E₀²
- (1/2)ε₀E₀²
- 2ε₀E₀²
Answer: (1/2)ε₀E₀²
Total instantaneous energy density u = ε₀E² = ε₀E₀² sin²(kx-ωt). Time average of sin² over a period is 1/2, so ⟨u⟩ = (1/2)ε₀E₀². This uses the principle that electric and magnetic contributions are equal, giving total u = ε₀E².
8. In an X-ray tube, electrons accelerated by 50 kV strike a tungsten target. What is the minimum wavelength of the emitted X-rays? (h = 6.63×10⁻³⁴ J s, c = 3×10⁸ m/s, e = 1.6×10⁻¹⁹ C)
- 0.100 nm
- 0.050 nm
- 0.0125 nm
- 0.025 nm
Answer: 0.025 nm
The minimum wavelength corresponds to the electron's full kinetic energy converting to a photon: λ_min = hc/(eV). Substituting values: λ_min = (6.63×10⁻³⁴ × 3×10⁸) / (1.6×10⁻¹⁹ × 50×10³) = 2.486×10⁻¹¹ m = 0.02486 nm ≈ 0.025 nm.
9. Which two of Maxwell's equations together predict the existence of electromagnetic waves?
- Faraday's law and Ampere-Maxwell law
- Gauss's law for electricity and Gauss's law for magnetism
- Gauss's law for electricity and Faraday's law
- Gauss's law for magnetism and Ampere-Maxwell law
Answer: Faraday's law and Ampere-Maxwell law
Faraday's law (changing magnetic field produces electric field) and Ampere-Maxwell law (changing electric field produces magnetic field) together show that a disturbance can propagate as a self-sustaining wave, even in vacuum.
10. Which of the following statements about ultraviolet (UV) radiation is correct?
- UV-A is completely absorbed by the ozone layer.
- UV-C is used for vitamin D synthesis in human skin.
- The ozone layer absorbs most of the Sun's UV-B and UV-C radiation.
- UV radiation has wavelengths longer than visible light.
Answer: The ozone layer absorbs most of the Sun's UV-B and UV-C radiation.
According to NCERT, the ozone layer in the stratosphere absorbs most of the Sun's UV-B and UV-C radiation, protecting life on Earth. UV-A (315-400 nm) reaches the ground, while UV-B and UV-C are largely absorbed.
11. Which of Maxwell's equations states that magnetic monopoles do not exist?
- ∮ E·dA = q/ε₀
- ∮ B·dl = μ₀ (I + ε₀ dΦ_E/dt)
- ∮ E·dl = -dΦ_B/dt
- ∮ B·dA = 0
Answer: ∮ B·dA = 0
Gauss's law for magnetism, ∮ B·dA = 0, states that the net magnetic flux through any closed surface is zero. This implies that there are no magnetic monopoles; magnetic field lines always form closed loops.
12. The magnetic field amplitude of an electromagnetic wave in vacuum is 2 × 10^{-7} T. What is the average energy density of the magnetic field?
- 7.96 × 10^{-9} J/m^3
- 3.18 × 10^{-8} J/m^3
- 6.37 × 10^{-8} J/m^3
- 1.59 × 10^{-8} J/m^3
Answer: 7.96 × 10^{-9} J/m^3
Average magnetic energy density is ⟨u_B⟩ = B₀²/(4μ₀). With B₀ = 2×10^{-7} T and μ₀ = 4π×10^{-7} T m/A, ⟨u_B⟩ = (4×10^{-14})/(4×4π×10^{-7}) = 10^{-14}/(16π×10^{-7}) = 10^{-7}/(16π) ≈ 7.96×10^{-9} J/m^3.