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Chapter 8: Electromagnetic Waves - Challenger (Level 3)
Section A: Displacement Current & Generalized Ampere's Law
1.
A parallel plate capacitor consists of circular plates of radius $a$ and separation $d$. If it is being charged by a steady current $i$, derive the expression for the magnetic field $B$ as a function of radial distance $r$ from the central axis for both $r \leq a$ and $r > a$.
2.
In Question 1, show that the magnetic field at the edge of the plate ($r=a$) calculated using the conduction current $i$ in the wires is identical to that calculated using the displacement current $i_d$ in the gap.
3.
The electric field between the plates of a parallel plate capacitor varies with time as $E = E_0 \sin(\omega t)$. Calculate the maximum displacement current density ($J_d$) in the gap.
4.
A time-varying electric field $E(t)$ is confined to a cylindrical region of radius $R$. Find the induced magnetic field at distance $r < R$ if $dE/dt = K$ (constant).
AI Image Prompt:
A high-quality 3D diagram showing two circular parallel plates of a capacitor. Between the plates, draw a cylindrical region showing a non-uniform electric field E represented by red vectors of varying lengths. Around this central axis, draw a circular Amperian loop of radius r. Show blue concentric magnetic field lines swirling around the changing E-field. The background should be fully white, landscape mode, and mathematically precise.
Filename: Level3_Q4_DisplacementVector.jpg
5.
Prove analytically that the displacement current in the gap of a capacitor is exactly equal to the conduction current in the connecting wires even when the dielectric medium has a relative permittivity $\epsilon_r$.
6.
A capacitor with capacitance $C$ is being charged by a battery of EMF $V$ through a resistor $R$. Derive the expression for the displacement current as a function of time $t$ after the switch is closed.
7.
An electron is moving with a constant acceleration $a$ along a straight line. Does it produce a displacement current in the surrounding space? Justify using Maxwell's logic.
8.
Show that the Ampere-Maxwell law can be written as $\oint \vec{B} \cdot d\vec{l} = \mu_0 \epsilon_0 \frac{d\Phi_E}{dt}$ for a region with zero conduction current.
9.
Calculate the displacement current if the charge on the capacitor plates is given by $q(t) = q_0 e^{-t/\tau}$.
10.
In a medium with conductivity $\sigma$ and permittivity $\epsilon$, find the ratio of conduction current density ($J_c$) to displacement current density ($J_d$) if the applied electric field is $E = E_0 \sin(\omega t)$.
Section B: Maxwell’s Equations & Wave Dynamics
11.
Derive the wave equation for the electric field in a vacuum starting from Maxwell's four equations in differential or integral form.
12.
Show that the speed of EM waves in a vacuum is exactly $1 / \sqrt{\mu_0 \epsilon_0}$ by comparing the derived wave equation with the standard form $\nabla^2 E = \frac{1}{v^2} \frac{\partial^2 E}{\partial t^2}$.
13.
Using Maxwell's equations, prove that for a plane EM wave, the electric field vector $\vec{E}$ and magnetic field vector $\vec{B}$ must be perpendicular to the direction of wave propagation.
14.
Show that the ratio of the magnitudes of $\vec{E}$ and $\vec{B}$ in a plane EM wave in free space is equal to the speed of light ($c$).
15.
If an EM wave enters a non-magnetic medium ($\mu_r = 1$) with refractive index $n$, how do the amplitudes $E_0$ and $B_0$ change compared to their values in a vacuum? Assume intensity remains constant.
16.
Derive the expression for the average energy density of an EM wave in terms of only the magnetic field amplitude $B_0$.
17.
A plane EM wave is given by $E_y = 60 \sin[2\pi \times 10^8 (t - x/c)]\text{ V/m}$. Calculate the energy contained in a cylinder of cross-section $10\text{ cm}^2$ and length $50\text{ cm}$ along the x-axis.
18.
State the physical significance of the Poynting Vector ($\vec{S} = \frac{1}{\mu_0} \vec{E} \times \vec{B}$). Calculate its time-average for a plane wave.
19.
Prove that the radiation pressure exerted by an EM wave of intensity $I$ on a perfectly reflecting surface is $2I/c$.
20.
A radio station transmits power $P$ isotropically. Derive the expression for the peak electric field $E_0$ at a distance $r$ from the station.
Section C: EM Wave Characteristics & Mixed Logic
21.
The electric field of a wave is $\vec{E} = E_0 \hat{j} \cos(kz - \omega t)$. Find the corresponding magnetic field vector $\vec{B}$.
22.
An EM wave has an intensity of $1.35\text{ kW/m}^2$. Calculate the total force exerted by this wave on a $2.0\text{ m}^2$ solar panel which is perfectly absorbing.
23.
In Question 22, if the panel is tilted at an angle of $30^\circ$ to the direction of propagation, how does the radiation pressure change?
24.
A $100\text{W}$ light bulb is $5\%$ efficient in converting electrical energy to visible light. Calculate the amplitude of the magnetic field at a distance of $2\text{ m}$ from the bulb.
25.
Show that the time-average of the energy density $\langle u \rangle$ can be expressed as $\frac{1}{2} \epsilon_0 E_0^2$.
26.
If the frequency of an EM wave is doubled while propagating in a vacuum, how are its speed, momentum, and radiation pressure affected?
27.
What is the ratio of electric force to magnetic force exerted by an EM wave on an electron moving with velocity $v = 0.1c$ perpendicular to the wave direction?
28.
A laser beam of power $10\text{ mW}$ is focused on a spot of diameter $10\text{ }\mu\text{m}$. Calculate the peak electric field at the focus point.
29.
If the wave in the diagram has a wavelength $\lambda = 500\text{ nm}$, find the wave number $k$ and the angular frequency $\omega$.
AI Image Prompt:
A Cartesian 3D axis showing an EM wave. The E-field oscillates in the XY plane and B-field in the XZ plane. The wave propagates along +X. Clearly label the peak amplitudes E0 and B0. Show the wavelength lambda. The background must be fully white, landscape, and high-quality vector style.
Filename: Level3_Q29_WaveParameters.jpg
30.
Explain why a static magnetic field cannot produce an EM wave, but a static charge being shaken (oscillated) can.
Section D: Electromagnetic Spectrum - Advanced Applications
31.
An EM wave has a photon energy of $2.48\text{ eV}$. To which part of the EM spectrum does it belong? (Take $h = 6.63 \times 10^{-34}\text{ J}\cdot\text{s}, 1\text{ eV} = 1.6 \times 10^{-19}\text{ J}$).
32.
Why is the frequency of an EM wave a more fundamental property than its wavelength when the wave transitions between different media?
33.
Identify the EM radiation produced by (a) transitions of inner shell electrons in heavy atoms, and (b) transitions of nucleons within a nucleus. Compare their energy scales.
34.
Calculate the range of frequencies for the "FM Radio" band if the wavelengths span from $2.8\text{ m}$ to $3.4\text{ m}$.
35.
Microwaves are used in Radar because they suffer less diffraction. Prove this using the concept that diffraction is proportional to $\lambda/D$ (where $D$ is antenna size).
36.
Why are "Greenhouse gases" specifically transparent to visible light but opaque to infrared radiation? Explain in terms of molecular vibration frequencies.
37.
Find the frequency of a Gamma ray photon having a wavelength of $10^{-13}\text{ m}$. Calculate its energy in MeV.
38.
Arrange based on decreasing order of "Penetrating Power": X-rays, Gamma rays, UV rays, Microwaves. Justify.
39.
A remote control uses infrared LEDs. If the pulse frequency is $38\text{ kHz}$ (carrier frequency), is this the same as the EM wave frequency? Explain the difference.
40.
Ultraviolet rays are used in "Fingerprint Detection" using fluorescence. Briefly explain the physics of why the emitted light is visible while the source is invisible.
Section E: Numerical Challenge (Calculated Value Type)
41.
Calculate the displacement current between the plates of a $100\text{ pF}$ capacitor when the potential is changing at $10^9\text{ V/s}$.
42.
The average energy density of an EM wave is $2.12 \times 10^{-12}\text{ J/m}^3$. Find the amplitude of the magnetic field $B_0$.
43.
A parallel plate capacitor is being charged at $0.5\text{ A}$. The plates are squares of side $10\text{ cm}$. Find the rate of change of electric field ($dE/dt$) between the plates.
44.
Find the wavelength of an EM wave in a medium where $v = 2 \times 10^8\text{ m/s}$ and frequency is $5 \times 10^{14}\text{ Hz}$.
45.
In an EM wave, $E_{rms} = 10\text{ V/m}$. Calculate the average intensity $I$.
46.
A plane EM wave is incident on a surface of area $0.5\text{ m}^2$. If the wave delivers $180\text{ J}$ of energy in $1\text{ minute}$, find the average radiation pressure on the surface (assuming absorption).
47.
What is the amplitude of the magnetic field in an EM wave if the electric field amplitude is $450\text{ V/m}$?
48.
If $\epsilon_r = 2$ and $\mu_r = 2$ for a medium, what is the speed of EM waves in it?
49.
Find the frequency of an EM wave whose wave number $k$ is $31.4\text{ m}^{-1}$.
50.
A charged particle oscillates with frequency $10^9\text{ Hz}$. What is the wavelength of the EM wave produced?
51.
A $20\text{W}$ laser beam is concentrated on $1\text{ mm}^2$ area. Find the average energy density.
52.
The peak value of magnetic field is $20\text{ nT}$. Find the peak electric field.
53.
Energy per unit volume in a wave is $u$. What is the momentum per unit volume?
54.
If the refractive index of a medium is $2.0$, find the ratio of speed of light in vacuum to speed in medium.
55.
A capacitor plate has area $0.1\text{ m}^2$ and displacement current is $2\text{ A}$. Find $dE/dt$.
56.
Determine the wavelength of the radiation emitted by an oscillating dipole of frequency $300\text{ MHz}$.
57.
Calculate the energy of a photon of wavelength $0.1\text{ nm}$ in Joules.
58.
Find the amplitude of the electric field at $10\text{ m}$ from a $1000\text{W}$ point source of EM waves.
59.
Relative permittivity of distilled water is $81$. Find the speed of EM waves in it (neglect magnetic effects).
60.
A surface of $1\text{ m}^2$ receives $6 \times 10^3\text{ J}$ of energy in $10\text{ s}$. Find the force if fully reflected.