Question

A plane electromagnetic wave propagating in the $x$-direction has a wavelength of $5.0\text{mm}$. The electric field is in the $y$-direction and its maximum magnitude is $30{\text{Vm}}^{-1}$. Write suitable equations for the electric and magnetic fields as a function of $x$ and $t$.

• A. $E=30\sin \left[400\pi \right(ct-x\left)\right]$
• B. $E=30\sin \left[100\pi \right(ct-x\left)\right]$
• C. $B=30\sin \left[100\pi \right(ct-x\left)\right]$
• D. $B={10}^{-7}\sin \left[400\pi \right(ct-x\left)\right]$

Answer 1

Answer: (A), (D)

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

Given:
$\lambda =5\times {10}^{-3}\text{m}$

$E_0=30\text{V/m}$

The equation for the electric and the magnetic fields in the form of wave may be written as

$E=E_0\sin \omega (t-\frac{x}{c})$

$B=B_0\sin \omega (t-\frac{x}{c})$

We have, $\omega =2\pi \nu =\frac{2\pi c}{\lambda }$

Thus, $E=E_0\sin \left[\frac{2\pi }{\lambda }\right(ct-x\left)\right]$

$\Rightarrow E=30\sin \left[\frac{2\pi }{5\times {10}^{-3}}\right(ct-x\left)\right]$

or, $E=30\sin \left[400\pi \right(ct-x\left)\right]$

The maximum value of magnetic field is given by

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

so, $B=B_0\sin \left[\frac{2\pi }{\lambda }\right(ct-x\left)\right]$

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

Hence, options $\left(A\right)$ and $\left(D\right)$ are the correct answers.