Example

Question:

The magnetic field in a plane electromagnetic wave is given by \( B_{y} = (2 \times 10^{-7})\,\text{T} \cdot \sin(0.5 \times 10^{3}x + 1.5 \times 10^{11}t) \).
(a) What is the wavelength and frequency of the wave?
(b) Write an expression for the electric field.

Solution:

(a) Compare to \( B_y = B_0 \sin\left(2\pi\left(\frac{x}{\lambda} + \frac{t}{T}\right)\right) \):
\[ \lambda = \frac{2\pi}{0.5 \times 10^3} = 1.26\,\text{cm} \] \[ \nu = \frac{1}{T} = \frac{1.5 \times 10^{11}}{2\pi} = 23.9\,\text{GHz} \] (b) \( E_0 = B_0 c = 2 \times 10^{-7} \times 3 \times 10^{8} = 60\,\text{V/m} \).
The electric field component along the \(z\)-axis is: \[ E_z = 60 \sin(0.5 \times 10^3 x + 1.5 \times 10^{11} t)\,\text{V/m} \]