Electromagnetic Waves Set-3

Q1. A plane electromagnetic wave propagates in a non-magnetic dielectric medium with relative permittivity $\epsilon_r=4$ and relative permeability $\mu_r=1$. What is its speed $v$ in the medium?

Q2. A monochromatic plane electromagnetic wave in vacuum has electric field amplitude $E_0=120\ \mathrm{V/m}$. Calculate the time‑averaged intensity $\langle S\rangle$ (use $\langle S\rangle=\tfrac{1}{2}\epsilon_0 c E_0^2$ if needed and $\epsilon_0=8.85\times10^{-12}\ \mathrm{F/m}$, $c=3.00\times10^8\ \mathrm{m/s}$).

Q3. At a fixed point in space the measured time dependence of fields is $E(t)=E_0\sin(\omega t)$ and $B(t)=B_0\cos(\omega t)$ with $E_0=c\,B_0$. Based on these traces, which statement about the time‑averaged Poynting vector $\langle\vec{S}\rangle$ is correct?

Q4. A plane wave in vacuum is normally incident on a lossless dielectric with relative permittivity $\epsilon_r=2.25$ and $\mu_r=1$. What fraction of the incident intensity is transmitted into the dielectric (neglect absorption)?

Q5. Assertion: A time‑varying electric field in vacuum can produce a magnetic field even where there is no conduction current.
Reason: Maxwell added the displacement current density $\epsilon_0\frac{\partial\vec{E}}{\partial t}$ to Ampère's law so that $\nabla\times\vec{B}=\mu_0\vec{J}+\mu_0\epsilon_0\frac{\partial\vec{E}}{\partial t}$; this term implies a changing $\vec{E}$ generates $\vec{B}$.

Q6. A plane electromagnetic wave of frequency $f=1.00\ \mathrm{GHz}$ is incident on copper with conductivity $\sigma=5.8\times10^7\ \mathrm{S/m}$ and $\mu=\mu_0$. Estimate the skin depth $\delta=\sqrt{\dfrac{2}{\omega\mu\sigma}}$ in copper (give answer in $\mu$m; use $\mu_0=4\pi\times10^{-7}\ \mathrm{H/m}$).

Q7. Assertion: A $90^\circ$ phase difference between the electric and magnetic fields of a plane electromagnetic wave implies the wave is circularly polarized.
Reason: Circular polarization is produced when two orthogonal components of the electric field have equal amplitude and a $90^\circ$ phase difference.

Q8. Two identical plane waves $E_1=E_0\sin(kx-\omega t)$ and $E_2=E_0\sin(kx+\omega t)$ superpose to give $E(x,t)=2E_0\sin(kx)\cos(\omega t)$. At $x=\lambda/8$ one finds the instantaneous Poynting vector $S(x,t)\propto\sin(2\omega t)$. Which statement about energy flow at this point is correct?

Q9. Unpolarized light of intensity $I_0$ passes through three ideal linear polarizers. The first is at $0^\circ$, the second at $45^\circ$ to the first, and the third at $90^\circ$ to the first. What intensity emerges after the third polarizer?

Q10. A plane electromagnetic wave in air ($n_1=1.00$) is incident on water ($n_2=1.33$). For non‑magnetic media ($\mu=\mu_0$), the Brewster angle satisfies $\theta_B=\arctan\!\left(\dfrac{n_2}{n_1}\right)$. What is $\theta_B$ (approximately)?