Electromagnetic Waves (Class 12 Physics) MCQ Set-4

Q1. A plane electromagnetic wave propagates in vacuum. A field-measuring instrument records the rms value of the electric field as $E_{\mathrm{rms}}=10\ \mathrm{V/m}$. What is the time-averaged intensity $\langle S\rangle$ (time-average of the magnitude of the Poynting vector) of the wave? Use $\epsilon_0=8.85\times10^{-12}\ \mathrm{F\,m^{-1}}$ and $c=3.00\times10^8\ \mathrm{m\,s^{-1}}$.

Q2. A monochromatic plane EM wave of angular frequency $\omega$ propagates in a non-magnetic dielectric (permittivity $\epsilon$, permeability $\mu=\mu_0$) with small conductivity $\sigma$. For a good dielectric ($\sigma/(\omega\epsilon)\ll1$) the attenuation constant is approximated by $\alpha\approx\dfrac{\sigma}{2}\sqrt{\dfrac{\mu}{\epsilon}}$. Show that the amplitude after one wavelength $\lambda$ in the medium is reduced approximately by $e^{-\alpha\lambda}\approx e^{-\pi\sigma/(\omega\epsilon)}$. If $\dfrac{\sigma}{\omega\epsilon}=0.10$, what is the numerical value of the amplitude ratio $E(z+\lambda)/E(z)$ after one wavelength?

Q3. In a dispersive medium the refractive index depends on angular frequency as $n(\omega)=n_0+\dfrac{A}{\omega^2}$ with constants $n_0>0$ and $A>0$. For frequencies satisfying $\omega^2>\dfrac{A}{n_0}$, the phase velocity is $v_p=\dfrac{c}{n(\omega)}$ and the group velocity is $v_g=\dfrac{c}{\,n(\omega)+\omega\,\dfrac{dn}{d\omega}\,}$. Which of the following correctly states the relation between $v_p$ and $v_g$ for those frequencies?

Q4. Two identical monochromatic plane electromagnetic waves of angular frequency $\omega$ and electric-field amplitude $E_0$ travel in opposite directions in vacuum and form a standing wave. Which statement correctly describes the time-averaged energy densities at a point which is an electric-field antinode (where $\cos kx=1$)?

Q5. Assertion (A): For an electromagnetic wave with angular frequency $\omega$ propagating in a collisionless plasma with plasma frequency $\omega_p$ and $\omega>\omega_p$, the phase velocity is $v_p=\dfrac{c}{\sqrt{1-(\omega_p/\omega)^2}}>c$.
Reason (R): This superluminal phase velocity does not violate special relativity because information and energy propagate at the group velocity $v_g=\dfrac{d\omega}{dk}=c\sqrt{1-(\omega_p/\omega)^2}<c$, and in fact $v_p\,v_g=c^2$.
Which option is correct?

Q6. A plane electromagnetic wave in vacuum is described by the electric field $\vec E(x,t)=E_0\sin(kx-\omega t)\,\hat y$ with amplitude $E_0=100\ \text{V m}^{-1}$. Using $\varepsilon_0=8.85\times10^{-12}\ \text{F m}^{-1}$ and $c=3.00\times10^8\ \text{m s}^{-1}$, the time-averaged intensity is $I_{\text{avg}}=\dfrac{1}{2}\varepsilon_0 c E_0^2$. Its numerical value (in $\text{W m}^{-2}$) is:

Q7. A plane electromagnetic wave in air ($n_1=1$) is normally incident on a non-magnetic dielectric ($\mu=\mu_0$) whose relative permittivity is $\varepsilon_r=4$. Neglect absorption and multiple internal reflections. The fraction of incident intensity transmitted across the first surface into the dielectric is:

Q8. A plane electromagnetic wave of amplitude $E_0$ in vacuum is normally incident on a non-magnetic dielectric ($\mu=\mu_0$) with relative permittivity $\varepsilon_r=4$. Neglect reflections beyond the first interface. If the transmitted electric amplitude is $E_t$, the time-averaged electromagnetic energy density in the dielectric is $u_t=\tfrac{1}{2}\varepsilon E_t^2$ and in vacuum $u_i=\tfrac{1}{2}\varepsilon_0 E_0^2$. The ratio $\dfrac{u_t}{u_i}$ equals:

Q9. Consider a parallel-plate capacitor being charged by a battery. Let the plates be large circular disks of radius $R$ separated by distance $d\ll R$. Assertion A: "During charging, the electromagnetic energy delivered by the battery flows into the capacitor through the region between the plates as a Poynting flux (fields carry energy into the gap)." Reason R: "The energy density stored between the plates at any instant is $u=\dfrac{1}{2}\varepsilon_0 E^2$, where $E$ is the electric field between the plates." Choose the correct option:

Q10. A monochromatic plane electromagnetic wave of intensity $I=1000\ \mathrm{W\,m^{-2}}$ is incident at angle $\theta=60^\circ$ on a flat surface with reflectivity $r=0.6$ (fraction of incident power reflected; remainder absorbed). The radiation pressure normal to the surface is $p=\dfrac{(1+r)I\cos^2\theta}{c}$. Using $c=3.00\times10^8\ \mathrm{m\,s^{-1}}$, the numerical value of $p$ is approximately:

Q11. A plane electromagnetic wave in vacuum has electric field amplitude $E_0=200\ \mathrm{V/m}$. Calculate the time‑averaged intensity (time‑average magnitude of the Poynting vector) of the wave. Use $\varepsilon_0=8.85\times10^{-12}\ \mathrm{F/m}$ and $c=3.00\times10^8\ \mathrm{m/s}$.

Q12. A plane electromagnetic wave in vacuum ($\mu=\mu_0$) is normally incident on a non‑magnetic dielectric with relative permittivity $\varepsilon_r=4$. Neglect absorption. Using the characteristic impedance $Z=\sqrt{\mu/\varepsilon}$, what fraction of the incident intensity is transmitted into the dielectric?

Q13. For a conductor with conductivity $\sigma=1.0\times10^7\ \mathrm{S/m}$, an electromagnetic wave of frequency $f=1.0\times10^8\ \mathrm{Hz}$ has skin depth $\delta=\sqrt{\dfrac{2}{\mu_0\sigma\omega}}$ where $\omega=2\pi f$. If a plane wave at this frequency traverses a slab whose thickness equals one skin depth ($d=\delta$), what fraction of the incident intensity emerges from the far side (time‑averaged transmitted intensity divided by incident intensity)?

Q14. Assertion (A): For an electromagnetic wave in a material with refractive index $n(\omega)$ the group velocity $v_g$ and phase velocity $v_p$ are related by $v_g=\dfrac{v_p}{1+\dfrac{\omega}{n}\dfrac{dn}{d\omega}}$.
Reason (R): Since $k(\omega)=\dfrac{n(\omega)\,\omega}{c}$, one gets $\dfrac{dk}{d\omega}=\dfrac{n+\omega\,\dfrac{dn}{d\omega}}{c}$ and hence $v_g=\dfrac{d\omega}{dk}=\dfrac{c}{\,n+\omega\,\dfrac{dn}{d\omega}\,}$.

Q15. A plane electromagnetic wave of intensity $I=1.00\times10^3\ \mathrm{W/m^2}$ is incident on a perfectly reflecting flat surface at angle $\theta=60^\circ$ measured from the surface normal. What is the radiation pressure (force per unit area normal to the surface) on the surface? Take $c=3.00\times10^8\ \mathrm{m/s}$.