An EM wave has momentum given by the equation: 𝒑 = 𝜇 𝑜
𝜖 𝑜
∭ 𝑺 𝑑
3 𝑥.
For the above and following we assume average values. It transfers momentum given by: 𝑑𝒑 𝑑𝑡
= 𝜇 𝑜
𝜖 𝑜 𝑑 𝑑𝑡
∭ 𝑺 𝑑
3 𝑥.
In a given time the length of the wave incident on an area A is given by c dt. We can thus rewrite the equation: 𝑑𝒑 𝑑𝑡
= 𝜇 𝑜
𝜖 𝑜 𝑑 𝑑𝑡
∫ 𝑐 𝑑𝑡 ∬ 𝑑𝑎 𝑺, which becomes, via the fundamental theorem of calculus: 𝑑𝒑 𝑑𝑡
= 𝜇 𝑜
𝜖 𝑜 𝑐 ∬ 𝑑𝑎 𝑺, 𝑑𝒑 𝑑𝑡
=
1 𝑐
∬ 𝑑𝑎 𝑺
Thus the pressure on an object is given by:
𝑃 =
𝑆 𝑐
= 𝑢 𝑒𝑚
, so we see the pressure is equal to the energy density in the EM waves!
Side note: the above assumes the waves are absorbed by an object. If they reflect, as off of a conductor, we find the change in momentum and pressure should be doubled: 𝑑𝒑 𝑑𝑡
=
2 𝑐
∬ 𝑑𝑎 𝑺,
𝑃 = 2
𝑆 𝑐
= 2 𝑢 𝑒𝑚
.