Electromagnetic Waves Set-1

Q1. At $t=0$ the $y$–component of the electric field of a plane monochromatic electromagnetic wave travelling in the $+x$ direction in vacuum is given by $E_y(x,0)=E_0\sin(kx)$ with $E_0>0$. For this travelling wave the magnetic field is $B_z(x,t)$. At the point $x=\pi/(2k)$ and $t=0$ which of the following is correct about the instantaneous electric and magnetic energy densities $u_E=\tfrac{1}{2}\varepsilon_0E^2$ and $u_B=\tfrac{1}{2\mu_0}B^2$?

Q2. A plane electromagnetic wave in vacuum is described by $\mathbf{E}(x,t)=120\sin(kx-\omega t)\,\hat{\jmath}\ \mathrm{V/m}$. Calculate the time–averaged intensity $I$ and the amplitude of the magnetic field $B_0$. (Use $c=3.0\times10^8\ \mathrm{m/s}$ and $\varepsilon_0=8.85\times10^{-12}\ \mathrm{F/m}$.)

Q3. Assertion (A): Two identical monochromatic plane waves travelling in opposite directions in vacuum form a standing electromagnetic wave in which instantaneous electric and magnetic energy densities are equal at every point and time.
Reason (R): For the standing wave $E_{\text{tot}}(x,t)=2E_0\sin(kx)\cos(\omega t)$ and $B_{\text{tot}}(x,t)=\dfrac{2E_0}{c}\cos(kx)\sin(\omega t)$; thus at a fixed point in space $E$ and $B$ oscillate $90^\circ$ out of phase in time, so instantaneous energy densities generally differ.

Q4. A monochromatic plane EM wave in vacuum with vacuum wavelength $\lambda_0=600\ \mathrm{nm}$ enters a non-magnetic dielectric ($\mu=\mu_0$) having refractive index $n=1.5$. Inside the dielectric the (measured) electric field amplitude is $E_0=45\ \mathrm{V/m}$. Find (i) the wavelength $\lambda$ inside the dielectric, (ii) the amplitude of the magnetic field $B_0$ inside the dielectric, and (iii) the instantaneous ratio $u_E/u_B$ of electric to magnetic energy densities there.

Q5. Two plane waves of equal amplitude $E_0$ and angular frequency $\omega$ travel along $+x$ in vacuum. Their electric fields are $\mathbf{E}_1=E_0\cos(kx-\omega t)\,\hat{\jmath}$ and $\mathbf{E}_2=E_0\cos\!\big(kx-\omega t+\tfrac{\pi}{2}\big)\,\hat{k}$. At a fixed $x$, which of the following best describes the tip of the resultant electric vector as time evolves?

Q6. Assertion (A): In a collisionless electron plasma with plasma frequency $\omega_p$, an electromagnetic wave with angular frequency $\omega<\omega_p$ is evanescent and cannot propagate through the plasma.
Reason (R): The dispersion relation is $\omega^2=\omega_p^2+c^2k^2$, so if $\omega<\omega_p$ the quantity $k^2$ is negative and $k$ is imaginary, producing evanescent behaviour.

Q7. A plane EM wave of frequency $f=1.0\times10^9\ \mathrm{Hz}$ is incident on a thick copper slab with conductivity $\sigma=5.8\times10^7\ \mathrm{S/m}$ and permeability $\mu=\mu_0$. Using the skin–depth formula $\displaystyle \delta=\sqrt{\frac{2}{\omega\mu\sigma}}$, compute the skin depth $\delta$ and the factor by which the amplitude of the wave is reduced after passing a thickness $3\delta$ (field amplitude attenuation factor $=e^{-3}$).

Q8. A plane electromagnetic wave of intensity $I_0$ is totally reflected by a perfect conductor, producing a standing wave region in front of the conductor. What is the time–averaged Poynting vector $\langle\mathbf{S}\rangle$ at points inside the standing wave region (away from the conductor surface)?

Q9. Assertion (A): For a plane electromagnetic wave in vacuum $\mathbf{E}=E_0\sin(kx-\omega t)\,\hat{\jmath}$, the corresponding magnetic field can be written as $\mathbf{B}=(E_0/c)\sin(kx-\omega t)\,\hat{k}$.
Reason (R): This is because in vacuum the magnetic field is parallel to the electric field and its magnitude equals $E/c$.

Q10. The dispersion relation for electromagnetic waves in a cold collisionless plasma is $\omega^2=\omega_p^2+c^2k^2$. For a wave–packet centred at $k_0=\dfrac{\omega_p}{\sqrt{3}\,c}$ compute the phase velocity $v_{\rm ph}=\omega/k$, the group velocity $v_g=d\omega/dk$ at $k_0$, and state whether energy/information can propagate faster than $c$ in this medium.