JEE Main 2024 — Electromagnetic Waves Question with Solution

To find the correct equation for the magnetic field of the plane electromagnetic wave given its parameters, we can use a couple of known relationships from electromagnetism.

Firstly, the wavelength ($\lambda$) of the wave is given as $4\mathrm{mm}=4\times 10^{-3}\mathrm{m}$. The speed of light (and all electromagnetic waves in vacuum) is $c=3\times 10^{8}m/s$. Using these values, we can find the frequency ($f$) of the wave using the relationship:

$c=\lambda f$

$f=\frac{c}{\lambda }=\frac{3\times 10^8}{4\times 10^{-3}}=75\times 10^9\mathrm{Hz}$

The angular frequency ($\omega$) which appears in wave equations is related to the frequency by $\omega =2\pi f$. However, in this context, what we need is the wave vector ($k$), which defines how the phase of the wave changes with space. The wave vector $k=\frac{2\pi }{\lambda }$. So, for this wave, $k=\frac{2\pi }{4\times 10^{-3}}=\frac{\pi }{2}\times 10^3{\mathrm{m}}^{-1}$.

Knowing that the electric field ($\mathbf{E}$) and magnetic field ($\mathbf{B}$) are related as $E=cB$ in a vacuum, where $E$ and $B$ are the magnitudes of the electric and magnetic fields, respectively, we can calculate the magnitude of the magnetic field using the provided maximum electric field magnitude ($E=60{\mathrm{Vm}}^{-1}$).

$B=\frac{E}{c}=\frac{60}{3\times 10^8}=2\times 10^{-7}\mathrm{T}$

Therefore, the wave equation for the magnetic field, considering it propagates in the $x$-direction and oscillates in a direction perpendicular to both the $x$-direction and the direction of the electric field (thus, in the $z$-direction if $\mathbf{E}$ is in the $y$-direction), is:

$\mathbf{B}=B\sin (kx-\omega t)\hat{\mathbf{k}}$

$\mathbf{B}=2\times 10^{-7}\sin \left(\frac{\pi }{2}\times 10^3(x-3\times 10^{8}t)\right)\hat{\mathbf{k}}\mathrm{T}$

This is represented by Option B:

${\mathrm{B}}_z=2\times 10^{-7}\sin \left[\frac{\pi }{2}\times 10^3\left(x-3\times 10^8\mathrm{t}\right)\right]\hat{\mathrm{k}}\mathrm{T}$

Therefore, the correct answer is Option B.