PH20014: Electromagnetism1: Dr Paul Snow

Poynting’s Theorem in Source Free Vacuum

Maxwell’s Equations: B

Et

(M3) and

DH J

t

(M4).

For a source free vacuum, 0 , 0 and 0J . Thus, as 0B H and 0D E ,

0

HE -

t

0

EH

t

.

We take a scalar product of E

with both sides of Maxwell 4 gives,

0

EE H E

t

.

For any vector field, a standard result says that (A B) B A A B

So, here letting A

= H

and B

= E

allows us to change the LHS:

0

EH E (H E) E .

t

(1)

But, 0

HE -

t

so, 0 0

H 1 (H H)H E H

2t t

The factor of ½ arises from linking H

Ht

and

t

H

2

. Thus, we can return to (1) and write

it as,

2 2

00

1E H

2 2

H E

t t

where we have changed the order of H

and E

and also simplified the term with E

t

.

Now, we rearrange the terms and take the volume integral of both sides. We get –

2 2

0 0

1 1

2 2v v

E H dv E H dvt

We can now apply the divergence theorem to simplify the left hand side and we obtain the

following equation – and assign the meaning of the terms as shown below,

2 2

0 0

1 1

2 2S v

E H dS E H dvt

= Total power leaving the volume through

the closed surface

Rate of decrease of the

energy stored in electric and magnetic

fields