Electromagnetic Waves Electromagnetic Waves Set-5

Q1. A plane monochromatic electromagnetic wave in vacuum is described by $\mathbf{E}(x,t)=E_0\sin(kx-\omega t)\,\hat{y}$ with amplitude $E_0=150\ \mathrm{V/m}$. Using $c=3.00\times10^8\ \mathrm{m/s}$ and $\varepsilon_0=8.85\times10^{-12}\ \mathrm{F/m}$, estimate the time-averaged intensity (magnitude of the time-averaged Poynting vector) of the wave.

Q2. A plane electromagnetic wave is normally incident from medium 1 to medium 2. Medium 1 has $\varepsilon_1=2\varepsilon_0,\ \mu_1=\mu_0$ and medium 2 has $\varepsilon_2=\varepsilon_0,\ \mu_2=4\mu_0$. Both media are non-absorbing. What fraction of the incident intensity is transmitted into medium 2?

Q3. A plane electromagnetic wave of frequency $f=10\ \mathrm{MHz}$ is incident on a copper conductor with $\sigma=5.8\times10^{7}\ \mathrm{S/m}$, $\mu=\mu_0$, $\varepsilon\approx\varepsilon_0$. Using the good-conductor approximation with $\displaystyle \delta=\sqrt{\frac{2}{\omega\mu\sigma}}$, estimate the skin depth $\delta$ and the ratio $\delta/\lambda_{\rm cond}$ where $\lambda_{\rm cond}$ is the wavelength inside the conductor.

Q4. Assertion (A): A plane electromagnetic wave of angular frequency $\omega$ incident from vacuum onto a collisionless plasma with plasma frequency $\omega_p$ is completely reflected when $\omega<\omega_p$ (no propagating energy into the plasma).
Reason (R): For $\omega<\omega_p$ the dielectric function $\displaystyle \varepsilon(\omega)=1-\frac{\omega_p^2}{\omega^2}$ becomes negative, so the wavevector $k=\omega\sqrt{\mu_0\varepsilon(\omega)}$ is purely imaginary, leading to evanescent decay inside the plasma.

Q5. A plane electromagnetic wave in vacuum is normally incident on a non-absorbing medium with relative permittivity $\varepsilon_r=1$ and relative permeability $\mu_r=4$. Let the amplitude of the incident electric field be $E_i$. Which of the following statements is correct?

Q6. A plane electromagnetic wave in vacuum has electric field amplitude $E_0=200\ \mathrm{V\,m^{-1}}$. Calculate the time‑averaged energy density of the wave.

Q7. A plane electromagnetic wave in air ($n_1=1$) is normally incident on a glass slab with refractive index $n_2=1.5$. Assuming no absorption, what fraction of the incident power is transmitted into the glass?

Q8. A plane electromagnetic wave of intensity $I=500\ \mathrm{W\,m^{-2}}$ is normally incident on a perfectly absorbing disc of area $5\ \mathrm{cm^{2}}$. Using $c=3.00\times10^{8}\ \mathrm{m\,s^{-1}}$, what is the average radiation force on the disc?

Q9. Assertion (A): When a linearly polarized plane electromagnetic wave in vacuum is normally incident on a non‑magnetic dielectric ($\mu=\mu_0$) with refractive index $n>1$, the amplitude of the magnetic field inside the dielectric is greater than its amplitude in vacuum.
Reason (R): The electric‑field amplitude inside the dielectric is $E_{\rm in}=\dfrac{2}{1+n}E_{\rm inc}$ while the phase velocity becomes $v=\dfrac{c}{n}$, hence $B_{\rm in}=E_{\rm in}/v$ can exceed $B_{\rm inc}=E_{\rm inc}/c$ for $n>1$.

Q10. In an electron plasma electromagnetic waves obey dispersion relation $\omega^{2}=\omega_{p}^{2}+c^{2}k^{2}$. For a wave with $\omega=\sqrt{2}\,\omega_{p}$, determine the phase velocity $v_{p}$ and group velocity $v_{g}$ (in terms of $c$).

Q11. A plane electromagnetic wave in vacuum has magnetic field amplitude $B_0=3.0\times10^{-6}\ \mathrm{T}$. Calculate the time-averaged intensity (magnitude of the Poynting vector) of the wave. Use $c=3.0\times10^{8}\ \mathrm{m/s}$ and $\epsilon_0=8.85\times10^{-12}\ \mathrm{F/m}$.

Q12. A plane electromagnetic wave with intensity $I_0=1.0\times10^{3}\ \mathrm{W/m^2}$ is incident on a perfectly reflecting flat mirror at an angle $\theta=60^\circ$ to the normal. The mirror has area $A=2.0\times10^{-2}\ \mathrm{m^2}$. Find the magnitude of the average force exerted on the mirror. Take $c=3.0\times10^{8}\ \mathrm{m/s}$ and assume specular, perfect reflection.

Q13. A plane electromagnetic wave in air ($n_1=1.00$) has intensity $I=8.00\times10^{2}\ \mathrm{W/m^2}$ and is normally incident on a glass surface ($n_2=1.50$). Using $\epsilon_0=8.85\times10^{-12}\ \mathrm{F/m}$ and $c=3.0\times10^{8}\ \mathrm{m/s}$, calculate the amplitude of the transmitted electric field just inside the glass. (Use Fresnel relations at normal incidence.)

Q14. A single-layer anti-reflection coating is deposited on glass ($n_g=1.50$) for light incident from air ($n_{\text{air}}=1.00$). For normal incidence and design wavelength $\lambda_0=600\ \mathrm{nm}$ the coating index is chosen to satisfy the ideal no-reflection condition $n_c=\sqrt{n_{\text{air}}n_g}$. Calculate the minimum coating thickness $d$ that yields zero reflectance at $\lambda_0$.

Q15. A plane electromagnetic wave $\mathbf{E}_i=E_0\cos(kx-\omega t)\,\hat{\mathbf{y}}$ is normally incident on a perfect conductor at $x=0$ and is reflected as $\mathbf{E}_r=-E_0\cos(kx+\omega t)\,\hat{\mathbf{y}}$. For $x>0$ the total fields are $\mathbf{E}=\mathbf{E}_i+\mathbf{E}_r$ and $\mathbf{B}=\mathbf{B}_i+\mathbf{B}_r$. Calculate the maximum instantaneous magnitude of the Poynting vector $|\mathbf{S}|$ at position $x=\lambda/8$; express your answer in terms of $E_0$, $\epsilon_0$ and $c$.