Question

An electromagnetic wave travelling in the x-direction has frequency of $2\times {10}^{14}Hz$ and electric field amplitude of 27 $V{m}^{-1}$. From the options given below, which one describes the magnetic field for this wave?

• A. $\overrightarrow{B}(x,t)=(3\times {10}^{-8}T)\hat{j}$ $sin\left[2\pi \right(1.5\times {10}^{-8}x-2\times {10}^{14}t\left)\right]$
• B. $\overrightarrow{B}(x,t)=(9\times {10}^{-8}T)\hat{j}$ $sin\left[\right(1.5\times {10}^{-6}x-2\times {10}^{14}t\left)\right]$
• C. $\overrightarrow{B}(x,t)=(9\times {10}^{-8}T)\hat{k}$ $sin\left[2\pi \right(1.5\times {10}^{-6}x-2\times {10}^{14}t\left)\right]$
• D. $\overrightarrow{B}(x,t)=(9\times {10}^{-8}T)\hat{i}$ $sin\left[2\pi \right(1.5\times {10}^{-8}x-2\times {10}^{14}t\left)\right]$

Answer 1

The correct option is C $\overrightarrow{B}(x,t)=(9\times {10}^{-8}T)\hat{k}$ $sin\left[2\pi \right(1.5\times {10}^{-6}x-2\times {10}^{14}t\left)\right]$

Using Maxwell's equation

$\frac{E_{\circ }}{B_{\circ }}=c$ where c is velocity of light, $E_{\circ }$ and $B_{\circ }$ are amplitudes of electric and magnetic field.

given $E_{\circ }=27N/C$

The amplitude of magnetic field is $B_{\circ }=\frac{27}{3\times {10}^8}=9\times {10}^{-8}$

frequency of wave is given as $f=2X{10}^{14}$ Hz, it's angular frequency is given as $\omega =2\pi f=2\pi \times 2X{10}^{14}$

The equation of magnetic field can be given by

$B_{\circ }sin(kx-\omega t))=B_{\circ }sin(2\pi (\frac{x}{\lambda }-ft))=9\times {10}^{-8}sin(2\pi (\frac{x}{\lambda }-2X{10}^{14}t)\left)\right(\hat{j}or\hat{k})$

The direction of electromagnetic wave is given as x axis, the direction of magnetic field will be perpendicular to it, hence option B and C can only be the answer, however, in option B the angular frequency is not as calculated above.