Question

The equation that represents magnetic field of a plane electromagnetic wave which is propagating along \(x\)-direction with wavelength \(10\,\text{mm}\) and maximum electric field \(60\,\text{V m}^{-1}\) in \(y\)-direction is \((c=\text{speed of light})\):

• A. \((6\times 10^{-7})\sin[0.2\pi(ct-x)]\hat{k}\,\text{tesla}\)
• B. \((2\times 10^{-7})\sin[200\pi(ct-x)]\hat{k}\,\text{tesla}\)
• C. \((2\times 10^{-7})\sin[200\pi(ct-x)]\hat{i}\,\text{tesla}\)
• D. \((6\times 10^{-7})\sin[0.2\pi(ct-x)]\hat{i}\,\text{tesla}\)

Hint:

In an electromagnetic wave, \(\vec{E}\), \(\vec{B}\), and direction of propagation are mutually perpendicular. Also, \[ E_0=cB_0 \] and \[ k=\frac{2\pi}{\lambda}. \]

Answer 1

The Correct Option is B

Step 1: Determine the magnetic field amplitude and wave vector.
\( B_0 = E_0/c = 60/(3\times10^8) = 2\times10^{-7}\,\text{T} \). Wave propagates along x, so \( k = 2\pi/\lambda = 2\pi/(10^{-2}) = 200\pi\,\text{m}^{-1} \).

Step 2: Determine the direction of B and write the equation.
Wave along \( +x \), E along \( +y \), so B must be along \( +z \) (i.e., \( \hat{k} \)) by \( \vec{E}\times\vec{B} \parallel \hat{x} \). \[ \vec{B} = (2\times10^{-7})\sin[200\pi(ct-x)]\hat{k}\,\text{T} \] \[ \boxed{(2\times10^{-7})\sin[200\pi(ct-x)]\hat{k}\,\text{tesla}} \]