Question 36: EM waves

Question

The electric field in a plane electromagnetic wave is given by :
$$ E_y=69 \sin \left[0.6 	imes 10^3 x-1.8 	imes 10^{11} tight] \mathrm{V} / \mathrm{m} . $$
The expression for magnetic field associated with this electromagnetic wave is $\_\_\_\_$ T.

Accepted Answer

In a vacuum or air, the amplitudes of the electric field ( $\mathrm{E}_0$ ) and magnetic field ( $\mathrm{B}_0$ ) are related by the speed of light (c) :
$$ c=\frac{E_0}{B_0} \Rightarrow B_0=\frac{E_0}{c} $$
From the given equation $\mathrm{E}_{\mathrm{y}}=69 \sin \left[0.6 	imes 10^3 \mathrm{x}-1.8 	imes 10^{11} \mathrm{t}ight], \mathrm{E}_0=69 \mathrm{~V} / \mathrm{m}, \omega$ (angular frequency) $= 1.8 	imes 10^{11} \mathrm{rad} / \mathrm{s}$ and k (wave number) $=0.6 	imes 10^3 \mathrm{rad} / \mathrm{m}$
So, the amplitude of the magnetic field is,
$$ B_0=\frac{69}{3 	imes 10^8}=23 	imes 10^{-8}=2.3 	imes 10^{-7} T $$
In electromagnetic waves which is a transverse waves consists of the electric field ( $\overrightarrow{\mathrm{E}}$ ), magnetic field ( $\overrightarrow{\mathrm{B}}$ ), and the direction of propagation ( $\overrightarrow{\mathrm{s}}$ ) are all mutually perpendicular.
The wave is traveling along the +x direction and the electric field is oscillating along the y -axis.
Using the direction vector $\hat{\mathrm{s}}=\widehat{\mathrm{E}} 	imes \widehat{\mathrm{B}}$ :
$$ \hat{\imath}=\hat{\jmath} 	imes \widehat{B} $$
For this to be true, the magnetic field must be along the $\mathrm{z}-\operatorname{axis}(\hat{\mathrm{k}})$, because $\hat{\mathrm{j}} 	imes \hat{\mathrm{k}}=\hat{\mathrm{i}}$.
In an electromagnetic wave in a vacuum, the electric and magnetic fields are in phase.
So, the phase of magnetic field vector is, $\left(0.6 	imes 10^3 \mathrm{x}-1.8 	imes 10^{11} \mathrm{t}ight)$
Therefore, the magnetic field vector is $\mathrm{B}_{\mathrm{z}}=2.3 	imes 10^{-7} \sin \left[0.6 	imes 10^3 \mathrm{x}-1.8 	imes 10^{11} \mathrm{t}ight]$ T.
Hence, the correct option is (A).

Answer

$B_z=2.3 	imes 10^{-7} \sin \left[0.6 	imes 10^3 x-1.8 	imes 10^{11} tight]$

Answer

$B_z=2.3 	imes 10^{-7} \sin \left[0.6 	imes 10^3 x+1.8 	imes 10^{11} tight]$

Answer

$B_y=2.3 	imes 10^{-7} \sin \left[0.6 	imes 10^3 x-1.8 	imes 10^{11} tight]$

Answer

$B_y=69 \sin \left[0.6 	imes 10^3 x+1.8 	imes 10^{11} tight]$

EM waves

The electric field in a plane electromagnetic wave is given by : $ E_y=69 \sin \left[0.6 \times 10^3 x-1.8 \times 10^{11} t\right] \mathrm{V} / \mathrm{m} . The expression for magnetic field associated with this electromagnetic wave is \_\_\_\_$ T.

Choose the correct answer: