Two cylindrical regions contain the same total amount of electromagnetic energy. The first cylinder has a length L and the electromagnetic wave inside it has an electric field amplitude E₀ . The second cylinder has a length L 2 and the electromagnetic wave inside it has an electric field amplitude 4E₀ . The ratio of the radius of the first cylinder to the radius of the second cylinder is

Step-by-step solution

The average energy density of an electromagnetic wave is given by u = 1 2 ₀ E₀^2 , where E₀ is the electric field amplitude. The total electromagnetic energy in a cylindrical region is the product of the energy density and the volume of the cylinder ( V = r^2 L ). For the first cylinder: U₁ = ( 1 2 ₀ E₀^2 )( r₁^2 L) For the second cylinder: U₂ = ( 1 2 ₀ (4E₀)^2 ) ( r₂^2 L 2 ) = ( 1 2 ₀ (16E₀^2) ) ( r₂^2 L 2 ) = ( 1 2 ₀ E₀^2 )(8 r₂^2 L) Since the total energy is the same in both cylinders ( U₁ = U₂ ): ( 1 2 ₀ E₀^2 )