The energy of an electromagnetic wave contained in a small volume oscillates with
Show 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.
- zero frequency
- half the frequency of the wave
- double the frequency of the wave
- the frequency of the wave
The Correct Option is C
Solution and Explanation
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:
\( \frac{1}{4} \varepsilon_0 E_0^2 (1 - \cos(2\omega t - 2kx)) \)
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