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The magnetic field in a plane electromagnetic wave is given by
${B_y} = \left( {2 \times {{10}^{ - 7}}} \right)\sin \left( {0.5 \times {{10}^3}x + 1.5 \times {{10}^{11}}t} \right)T$
$(a)$ What is the wavelength and frequency of the wave?
$(b)$ Write an expression for the electric field.
Solution
$(a)$ Comparing the given equation with
$B_{y}=B_{0} \sin \left[2 \pi\left(\frac{x}{\lambda}+\frac{t}{T}\right)\right]$
We get, $\lambda=\frac{2 \pi}{0.5 \times 10^{3}} m =1.26 \,m$
and $\quad \frac{1}{T}=v=\left(1.5 \times 10^{11}\right) / 2 \pi=23.9 \,GHz$
$(b)$ $E_{0}=B_{0} c=2 \times 10^{-7} \,T \times 3 \times 10^{8} \,m / s =6 \times 10^{1}\, V / m$
The electric field component is perpendicular to the direction of propagation and the direction of magnetic field. Therefore, the electric field component along the $z$ -axis is obtained as
$E_{z}=60 \sin \left(0.5 \times 10^{3} x+1.5 \times 10^{11} t\right)\, V / m$
