Question

In a non-magnetic medium the equation of EM-wave propagation is $\vec{E} = 4 \sin(2\pi \times 10^7t - 0.8z)\hat{k}$ V/m [$\mu_0$ = 4$\pi$ x 10$^{-7}$ H/m; C (speed of light in vacuum) = 3 x 10$^8$ m/s] The average intensity of the given EM-wave is $I = \frac{N}{\pi}$W/m$^2$. Find the value of N.

Answer 1

4/(5π)

Answer 2

The problem asks us to calculate the average intensity of a given electromagnetic wave propagating in a non-magnetic medium.

The given electric field equation for the EM wave is: $\vec{E} = 4 \sin(2\pi \times 10^7t - 0.8z)\hat{k}$ V/m

Comparing this with the general form of a plane electromagnetic wave $E = E_0 \sin(\omega t - kz)$, we can identify the following parameters:

1. Peak electric field amplitude, $E_0 = 4$ V/m
2. Angular frequency, $\omega = 2\pi \times 10^7$ rad/s
3. Wave number, $k = 0.8$ rad/m

The speed of the electromagnetic wave in the medium ($v$) is given by the ratio of the angular frequency to the wave number: $v = \frac{\omega}{k}$ $v = \frac{2\pi \times 10^7 \text{ rad/s}}{0.8 \text{ rad/m}}$ $v = \frac{2\pi \times 10^7}{4/5} = \frac{10\pi \times 10^7}{4} = 2.5\pi \times 10^7$ m/s

The average intensity ($I_{avg}$) of an electromagnetic wave in a medium is given by the formula: $I_{avg} = \frac{E_0^2}{2\mu v}$ For a non-magnetic medium, the permeability ($\mu$) is approximately equal to the permeability of free space ($\mu_0$). So, $\mu = \mu_0 = 4\pi \times 10^{-7}$ H/m.

Now, substitute the values of $E_0$, $\mu_0$, and $v$ into the intensity formula: $I_{avg} = \frac{(4 \text{ V/m})^2}{2 \times (4\pi \times 10^{-7} \text{ H/m}) \times (2.5\pi \times 10^7 \text{ m/s})}$ $I_{avg} = \frac{16}{2 \times 4\pi \times 2.5\pi \times 10^{-7} \times 10^7}$ $I_{avg} = \frac{16}{2 \times 4\pi \times 2.5\pi}$ $I_{avg} = \frac{16}{8\pi \times 2.5\pi}$ $I_{avg} = \frac{16}{20\pi^2}$ $I_{avg} = \frac{4}{5\pi^2}$ W/m$^2$

The problem states that the average intensity is $I = \frac{N}{\pi}$ W/m$^2$. We have calculated $I_{avg} = \frac{4}{5\pi^2}$ W/m$^2$. Equating the two expressions for intensity: $\frac{N}{\pi} = \frac{4}{5\pi^2}$ To find $N$, multiply both sides by $\pi$: $N = \frac{4}{5\pi}$