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 u
E
W
E


1
2
 o
E
2

1
2
 o
2
E dV


Magnetic fields also store energy
 u
B

2
1
 o
B
2
W
B
 
2
1
 o
2
B dV
Total energy stored by electromagnetic fields per unit volume is… u

1
2
(
 o
E
2 
1
 o
B
2
)

Work done moving a charge…

Use the Lorentz formula:
F

(
 
)

The work in time-interval dt is then: dW F dl ( 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 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
   o
J
   o o

E
 t
And the identity

( E B ) B ( E ) E ( B )

The “Work-Energy” Theorem for
EM Fields…

Poynting’s Theorem tells us: dW
  d dt dt
 1
2
(
 o
E
2 
1
 o
2
)

1
 o

(

Change in energy stored in the fields
Energy radiated across surface by the electromagnetic fields

The Poynting Vector
S

1
 o
( )