Energy and Momentum of Electromagnetic Waves

The energy per unit volume stored in an electromagnetic field is

            U = 1/2(ε₀ E² + 1/μ₀ B²)                            … (1)

In the case of a monochromatic plane wave,
            B² = 1/c² E² = μ₀ ε₀ E²                             … (2)
. :            U = 1/2(ε₀ E² + 1/μ₀ μ₀ ε₀ E²)

If E points in the y-direction, then
            E(x, t) = E₀ cos (kx – ωt + δ) j
            B(x, t) = 1/2 E₀ cos (kx – ωt + δ) j
             U = ε₀ E² = ε₀ E²₀ cos² (kx – ωt + δ)                    … (3)

The average of cos2 over a complete cycle is 1/2.
. :         <U> = 1/2 ε₀ E²₀                                     … (4)
As the wave propagates, it carries this energy along with it.

The rate at which energy is transmitted through unit area perpendicular to the direction of propagation of energy is called pointing vector and is represented by S.

        S = 1/μ₀ (Ex B)                                    … (5)

For monochromatic place waves,
        S = c ε₀  E²₀ cos² (kx – ωt) + δ)I = cUi                        … (6)
The Poynting vector S is c times the energy density (U) of the field.

The average of cos² over a complete cycle is 1/2.
. :        <S> = 1/2 ε⁰ E²₀i = c <U>i
The average power per unit area transported by an electromagnetic wave,
        ρ = 1/c² S
For monochromatic plane waves,
        ρ = 1/c ε₀ E²₀ cos² (kx – ωt + δ)I = 1/cUi
    . :    U = ρc

The average of cos² over a complete cycle is 1/2.
        <ρ> = 1/2 1/c ε₀ E²₀i = 1/c <U>i


For more help in Energy and Momentum of Electromagnetic Waves please click the button below to submit your homework assignment.