(a) Use direct integration to show that a radiating charge q emits energy at the rate

In this expression, S is the Poynting vector, β = v/c, a = v̇, g = 1 − n̂ · β, and γ = 1/ √1 − β². The integration is carried out over the surface A of an enormously large sphere.

(b) Use direct integration to show that a radiating charge q emits linear momentum at the rate

In this expression, T_ij = ϵ₀ {E_iE_j + c2B_iB_j − 1/2 δ_ij (E² + B²)} is the electromagnetic stress tensor.

and use successive differentiation to derive other integrals you need.

dUEM dt = A dA gS = 276 9 - [la| - la B]. 3c 4 0