Transcript Poynting`s Theorem

Poynting’s Theorem
John Henry Poynting (1852-1914)
… energy conservation
To recap…

The energy stored in an electric field E is
expressed as the work needed to “assemble” a
group of point charges
uE   o E
1
2
2
WE    o E dV
1
2
2

Magnetic fields also store energy
uB 
1
2 o
B
2
WB   2 o B dV
1

2
Total energy stored by electromagnetic fields per unit
volume is…
1
1 2
2
u  ( o E 
B )
2
o
Work done moving a charge…

Use the Lorentz formula:
F  q ( E  v  B)

The work in time-interval dt is then:
dW  F  dl  q( E  v  B)  dl  qE  vdt
(remember: Magnetic fields do no work!)

Remember current density J! The moving
charge or charges constitute a current
density, so we can write
qE  vdt  E  J
which means the work can now be
expressed as
dW
  ( E  J )dV
dt
Remember that this is a rate of change of the energy (work) and so
represents power delivered per unit volume
Now use Maxwell’s Equations
E
  B   o J   o o
t
And the identity
 ( E  B)  B  ( E )  E  ( B)
The “Work-Energy” Theorem for
EM Fields…

Poynting’s Theorem tells us:
dW
d 1
1 2
1
2
   ( o E 
B )dV   ( E  B)dA
dt
dt 2
o
o
Change in energy stored in
the fields
Energy radiated across
surface by the
electromagnetic fields
The Poynting Vector
S
1
o
( E  B)