Question

The energy of an electromagnetic wave contained in a small volume oscillates with

• A. zero frequency
• B. half the frequency of the wave
• C. double the frequency of the wave
• D. the frequency of the wave

Hint:

The energy density of an EM wave is proportional to the square of the electric f ield. Squaring a sinusoidal function doubles the frequency.

Answer 1

The Correct Option is C

Step 1: Formula for Energy Density 

The energy density of the wave is given by:

\( \text{Energy density} = \frac{1}{2} \varepsilon_0 E_{\text{net}}^2 \)

Substitute \( E_{\text{net}} = E_0 \sin(\omega t - kx) \):

\[ \text{Energy density} = \frac{1}{2} \varepsilon_0 E_0^2 \sin^2(\omega t - kx) \]

Step 2: Use the Trigonometric Identity

Using the trigonometric identity \( \sin^2 x = \frac{1}{2}(1 - \cos 2x) \):

\[ \sin^2(\omega t - kx) = \frac{1}{2}(1 - \cos(2\omega t - 2kx)) \]

Substitute this into the energy density formula:

\[ \text{Energy density} = \frac{1}{2} \varepsilon_0 E_0^2 \cdot \frac{1}{2}(1 - \cos(2\omega t - 2kx)) \]

Simplify the expression:

\[ \text{Energy density} = \frac{1}{4} \varepsilon_0 E_0^2 (1 - \cos(2\omega t - 2kx)) \]

Final Answer:

The energy density of the wave is:

\( \frac{1}{4} \varepsilon_0 E_0^2 (1 - \cos(2\omega t - 2kx)) \)