Question

Given that the average intensity of a solar electromagnetic wave that hits our
atmosphere is 1367 W/m2

(a) what is the amplitude E0 of the electric field in this
wave, and

(b) what is its corresponding root-mean-square electric field, Erms?

Answer 1

Answer 2

Calculation:

(a) Amplitude of the Electric Field $E_0$$E_0$:

The intensity $I$ of an electromagnetic wave is related to the amplitude of the electric field $E_0$ by the formula:

$$I=\frac{1}{2}c{ϵ}_0{E}_0^2$$

Where:

• $I=1367{\text{W/m}}^2$ (given intensity),

• $c=3\times 10^8\text{m/s}$ (speed of light),

• ${ϵ}_0=8.854\times 10^{-12}\text{F/m}$ (permittivity of free space).

Step 1: Rearrange the formula to solve for $E_0$:

$$E_0=\sqrt{\frac{2I}{c{ϵ}_0}}$$

Step 2: Substitute the known values:

$$E_0=\sqrt{\frac{2\times 1367}{3\times 10^8\times 8.854\times 10^{-12}}}$$

Step 3: Simplify the denominator:

$$3\times 10^8\times 8.854\times 10^{-12}=2.6562\times 10^{-3}$$

Step 4: Calculate the fraction inside the square root:

$$\frac{2\times 1367}{2.6562\times 10^{-3}}\approx \frac{2734}{2.6562\times 10^{-3}}\approx 1.029\times 10^6$$

Step 5: Take the square root to find $E_0$:

$$E_0=\sqrt{1.029\times 10^6}\approx 1013\text{V/m}$$

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(b) Root-Mean-Square Electric Field $E_{\text{rms}}$$E_{\text{rms}}$:

The root-mean-square electric field is related to the amplitude $E_0$ by:

$$E_{\text{rms}}=\frac{E_0}{\sqrt{2}}$$

Step 1: Substitute $E_0=1013\text{V/m}$:

$$E_{\text{rms}}=\frac{1013}{\sqrt{2}}\approx \frac{1013}{1.414}\approx 716.5\text{V/m}$$

 Answer :
(a) The amplitude $E_0$ of the electric field is 1013 V/m.
(b) The root-mean-square electric field $E_{\text{rms}}$ is 716.5 V/m.