Question

A plane electromagnetic wave is propagating in the $x-$direction, has a wavelength of $5\text{mm}$. The electric field is in the $y-$direction and its maximum magnitude is $30\text{V/m}$. The possible equation for the associated magnetic field is -

• A. $\overrightarrow{B}={10}^{-7}\sin \left[400\pi \right(ct-x\left)\right]\hat{k}$
• B. $\overrightarrow{B}=30\sin \left[200\pi \right(ct-x\left)\right]\hat{j}$
• C. $\overrightarrow{B}={10}^{-7}\sin \left[200\pi \right(ct-x\left)\right]\hat{k}$
• D. $\overrightarrow{B}={10}^{-7}\sin \left[400\pi \right(ct-x\left)\right]\hat{i}$

Answer 1

The correct option is A $\overrightarrow{B}={10}^{-7}\sin \left[400\pi \right(ct-x\left)\right]\hat{k}$

We know that,

$c=\frac{E_0}{B_0}$

$\Rightarrow B_0=\frac{E_0}{c}=\frac{30}{3\times {10}^8}={10}^{-7}\text{T}$

Further,

$k=\frac{2\pi }{\lambda }=\frac{2\pi }{5\times {10}^{-3}}=400\pi {\text{m}}^{-1}$

Also, $c=\frac{\omega }{k}$

$\Rightarrow \omega =ck=400\pi c$

Now, the possible equation of magnetic field is,

$B=B_0\sin (\omega t-kx)$

$\Rightarrow B={10}^{-7}\sin (400\pi ct-400\pi x)$

$\Rightarrow B={10}^{-7}\sin \left[400\pi \right(ct-x\left)\right]$

As wave is travelling along $x-$direction and electric field is along $y-$direction, therefore magnetic field should be along $z-$ direction from $\hat{c}=\hat{E}\times \hat{B}$.

$\therefore \overrightarrow{B}={10}^{-7}\sin \left[400\pi \right(ct-x\left)\right]\hat{k}$

Hence, option $\left(A\right)$ is the correct answer.