A cubical region of side a contains an electromagnetic wave with an electric field amplitude E₀ . A second cubical region of side 2a contains a different electromagnetic wave but holds the exact same total electromagnetic energy as the first region. If c is the speed of light in vacuum, the magnetic field amplitude of the second wave is

Step-by-step solution

The average energy density of an electromagnetic wave can be expressed in terms of the electric field amplitude as u_E = 1 2 ₀ E₀^2 or in terms of the magnetic field amplitude as u_B = B₀^2 2 ₀ . The total energy in the first cubical region (volume V₁ = a^3 ) is: U₁ = ( 1 2 ₀ E₀^2 ) a^3 The total energy in the second cubical region (volume V₂ = (2a)^3 = 8a^3 ) is: U₂ = ( B₀'^2 2 ₀ ) (8a^3) Given that the total energies are equal ( U₁ = U₂ ): 1 2 ₀ E₀^2 a^3 = B₀'^2 2 ₀ (8a^3) Cancelling a^3 and rearranging for B₀'^2