Energy Derivation

We assume that the electric and magnetic energy density are

We = 1ϵE2 2

1 2Wm = 2μH

where E and H are the electric and magnetic fields, and ϵ and μ are just constants.

We believe the change of the electromagnetic field causes the electromagnetic energy change. This happens when either the electromagnetic energy exchanges with the environment, or the electromagnetic energy changes to another kind of energy. According to the experiment the energy goes through the surface of the system and can be described by:

S = E × H

PIC

On the other hand we found that only electric fields can do work. We assume during the unit time dt, and unit volume dV , the electric energy kE2dV dt changes to another kind of energy. So we obtain:

∫ ∮ ∫- -∂ (We +Wm )dV = S⋅da + kE2dV ∂t

This in turn can yield:

∂ ( 1 2 1 2)∇ ⋅(E ×H ) + kE ⋅E + ∂t 2ϵE + 2μH = 0

Further more we have:

( ) ( ) ∇ × E + μ ∂H-- ⋅H + - ∇ × H + kE + ϵ∂E ⋅E = 0 ∂t ∂t

This means:

∇ ×E +μ ∂H--= 0 ∂t

∂E- ∇ × H + kE + ϵ-∂t = 0

We define: D = ϵE, B = μH, and J = kE, So we can rewrite the last two equations:

∇ × E = - ∂B- ∂t

∂D-∇ × H = J + ∂t

Again using the continuity equation:

∂ρ∇ ⋅J + ---= 0 ∂t

which yields :

∇ ⋅D = ρ.

And finally, we retain the absence of free Magnetic monopoles so:

∇ ⋅B = ρm = 0

Back Home