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

Solution

Physics sir · 4 steps

Energy goes as the square of the field, and squaring a sine doubles how fast it oscillates.

1. Energy density. The energy stored per unit volume in an EM wave is u = 1/2ε₀ E² + B²/2μ₀ = ε₀ E², since the electric and magnetic parts are equal. So the energy depends on E², not on E.
2. Square the field. At a fixed point the field is E = E₀ sin ω t. Squaring it and using sin² θ = 1/2(1 - cos 2θ) gives the result below.

   u = ε₀ E₀² sin² ω t = 1/2ε₀ E₀²(1 - cos 2ω t)

3. Read the frequency. The changing part is cos 2ω t, which has angular frequency 2ω. The energy reaches a maximum at both the crest and the trough of the field, so it peaks twice in every wave cycle.
4. So. The energy in a small volume oscillates at double the frequency of the wave, which is option (a).

The idea to keep. Any quantity that goes as the square of a sinusoid, like energy or power, oscillates at twice the original frequency. Its average stays constant.

Answer from the NTA final answer key. Solution by PrepWiser.