For a sinusoidal plane electromagnetic wave in vacuum, let uₘₐₓ be the maximum total electromagnetic energy density and ⟨u⟩ be the time-averaged total energy density. If uₘₐₓ/⟨u⟩ = N, find N.
Physics · Electromagnetic wave · displacement current and properties of electromagnetic wave
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Question

For a sinusoidal plane electromagnetic wave in vacuum, let uₘₐₓ be the maximum total electromagnetic energy density and ⟨u⟩ be the time-averaged total energy density. If uₘₐₓ/⟨u⟩ = N, find N.

Physics · Electromagnetic wave · displacement current and properties of electromagnetic wave

G1 Physics · Accepted Answer

For a sinusoidal wave, E = E₀ cosφ. The total instantaneous electromagnetic energy density is u = ε₀E² = ε₀E₀²cos²φ. Therefore, uₘₐₓ = ε₀E₀². Since ⟨cos²φ⟩ = 1/2, the average total energy density is ⟨u⟩ = ½ε₀E₀². Hence, uₘₐₓ/⟨u⟩ = 2. Therefore N = 2.