Electromagnetic Waves Set-2

Q1. A plane electromagnetic wave in vacuum has electric field amplitude $E_0=300\ \mathrm{V\,m^{-1}}$. Calculate the time-averaged Poynting flux (intensity) $\langle S\rangle$ carried by this wave. Use $\mu_0=4\pi\times10^{-7}\ \mathrm{H\,m^{-1}}$ and $c=3.0\times10^8\ \mathrm{m\,s^{-1}}$.

Q2. Assertion (A): In a good conductor the amplitude of an electromagnetic wave decays exponentially with a penetration depth (skin depth) that is independent of frequency. Reason (R): The skin depth in a conductor is given by $\delta=\sqrt{\dfrac{2}{\omega\mu\sigma}}$, so it depends on angular frequency $\omega$.

Q3. A plane electromagnetic wave in vacuum has magnetic field amplitude $B_0=2.0\times10^{-6}\ \mathrm{T}$. What is the time-averaged total energy density $\langle u\rangle$ (electric + magnetic) in the wave? Use $\mu_0=4\pi\times10^{-7}\ \mathrm{H\,m^{-1}}$.

Q4. Inside a conductor the intensity of an electromagnetic wave decays as $I(x)=I_0\exp(-2x/\delta)$, where $\delta$ is the skin depth. A graph of $\ln(I/I_0)$ vs.\ $x$ (in metres) is a straight line with slope $-200\ \mathrm{m^{-1}}$. From this graph determine the skin depth $\delta$.

Q5. Monochromatic light has wavelength $600\ \mathrm{nm}$ in vacuum. It enters a glass slab with refractive index $n=1.50$. What is its wavelength inside the glass? (Assume the medium is non-dispersive at this frequency.)

Q6. A linearly polarized plane wave $\mathbf{E}=E_0\hat{x}\cos(kz-\omega t)$ propagates in $+z$ and is normally incident on an ideal conductor occupying the plane $z=0$. The incident and reflected waves form a standing wave for $z>0$. Which statement correctly describes the fields at the conductor surface $z=0$?

Q7. In a cold electron plasma the dispersion relation is $\omega^2=\omega_p^2+c^2k^2$. For $\omega=2.0\times10^{11}\ \mathrm{rad\,s^{-1}}$ and $\omega_p=1.0\times10^{11}\ \mathrm{rad\,s^{-1}}$, calculate the group velocity $v_g=d\omega/dk$ of the wave. Take $c=3.0\times10^8\ \mathrm{m\,s^{-1}}$.

Q8. A medium shows these local behaviours of refractive index $n(\omega)$ around a frequency $\omega_0$:

• Region I: $n$ increases linearly from $1.40$ at $\omega_0-\Delta\omega$ to $1.60$ at $\omega_0+\Delta\omega$.
• Region II: $n$ decreases linearly from $1.60$ at $\omega_0-\Delta\omega$ to $1.50$ at $\omega_0+\Delta\omega$.
  Assuming $\Delta\omega$ and $\omega_0$ are the same in both cases, at which region will the group velocity $v_g=\dfrac{c}{\,n+\omega\,dn/d\omega\,}$ at $\omega_0$ be larger?

Q9. Assertion (A): In certain frequency ranges of a dispersive medium the phase velocity of electromagnetic waves can exceed the speed of light in vacuum $c$. Reason (R): Phase velocity exceeding $c$ does not violate special relativity because phase velocity does not represent the velocity of information or energy transfer; the signal or front velocity remains $\le c$.

Q10. An electromagnetic plane wave with frequency $\omega<\omega_p$ (below the plasma frequency) produces evanescent fields inside a plasma and decays exponentially with depth. Which statement about the time-averaged Poynting vector $\langle\mathbf{S}\rangle$ inside the plasma is correct?