Question

Sunlight just outside Earth’s atmosphere has an intensity of$\mathbf{1}\mathbf{.}\mathbf{40}\mathbf{}{\mathbf{\text{kW/m}}}^{\mathbf{\text{2}}}$. Calculate${\mathit{E}}_{\mathbf{m}}$ (a) and (b)${\mathit{B}}_{\mathbf{m}}$for sunlight there, assuming it to be a plane wave.

Short answer

1. The amplitude of the electric field component is $1.03\times {10}^3\frac{\mathrm{V}}{\mathrm{m}}$.
2. The amplitude of the magnetic field component is $3.43\times {10}^{-6}\mathrm{T}$.

Step-by-step solution

Listing the given quantities

The intensity of sunlight $I=1.40{\text{kW/m}}^{\text{2}}=1.40\times {10}^3{\text{W/m}}^{\text{2}}$.

Understanding the concepts of amplitude

We use the intensity relation of the wave, square both sides, and solve for the electric field component. Finally, by substitutingtheelectric field component value inthemagnetic field component relation, we can find its amplitude.

(a) Calculations of the maximum value of the electric field component

The time-averaged rate per unit area at which energy is transported $S_{avg},$ is called the intensityI of the wave.

The maximum value of the electric component can be written as,

$E_m=\sqrt{2{\mu }_{0}Ic}$

Here ${\mu }_0$ is the permeability of the free space, and c is the speed of the light.

Substitute the values in the above expression, and we get,

$\begin{array}{rcl}{E}_m& =& \sqrt{2\left(4\pi \times {10}^{-7}\frac{\text{T}·\text{m}}{\text{A}}\right)\left(1.40\times {10}^3\frac{\text{W}}{{\text{m}}^{\text{2}}}\right)\left(3\times {10}^8\frac{\text{m}}{\text{s}}\right)}\\ & =& \sqrt{2\left(4\pi \times {10}^{-7}\right)\left(1.40\times {10}^3\right)\left(3\times {10}^8\right)\left(1\frac{\text{T}·\text{m}}{\text{A}}\times 1\frac{\text{W}}{{\text{m}}^{\text{2}}}\times 1\frac{\text{m}}{\text{s}}\times \frac{1\text{V}·\text{A}}{1\text{W}}\times \frac{\text{1}\text{V}·{\text{s/m}}^{\text{2}}}{1\text{T}}\right)}\\ & =& 1.03\times {10}^3\text{V/m}\end{array}$

The amplitude of the electric field component is $1.03\times {10}^3\frac{\mathrm{V}}{\mathrm{m}}$.

(b) Calculations of the amplitude of the magnetic field component Bm

The amplitude of the magnetic field component in the wave is given by,

$B_m=\frac{E_m}{c}$

Substitute the values in the above expression, and we get,

$\begin{array}{rcl}{B}_m& =& \frac{1.03\times {10}^3\frac{\text{V}}{\text{m}}}{3\times {10}^8\frac{\text{m}}{\text{s}}}\\ & =& 3.43\times {10}^{-6}\text{T}\end{array}$

The amplitude of the magnetic field component $B_m$is $3.43\times {10}^{-6}\text{T}$.