Next: Electromagnetic radiation
Up: Electromagnetic energy and momentum
Previous: Electromagnetic momentum
Momentum conservation
It follows from Eqs. (1036) and (1050) that the momentum
density of electromagnetic fields can be written

(1058) 
Now, a momentum conservation equation for electromagnetic fields
should take the integral form

(1059) 
Here, and run from 1 to 3 (1 corresponds to the direction,
2 to the direction, and 3 to the direction). Moreover, the Einstein
summation convention is employed for repeated indices (e.g.,
). Furthermore, the tensor
represents the flux of the th component of electromagnetic momentum in the direction. This tensor (a tensor is a direct generalization of a vector with two indices instead of one) is called
the momentum flux density. Hence, the above equation states
that the rate of loss of electromagnetic momentum in some volume
is equal to the flux of electromagnetic momentum across the bounding surface plus the rate at which momentum is transferred to matter inside .
The latter rate is, of course, just the net electromagnetic force acting
on matter inside : i.e., the volume integral of the electromagnetic
force density,
.
Now, a direct generalization of the divergence theorem states that

(1060) 
where , , etc. Hence, in differential form, our
momentum conservation equation for electromagnetic fields is written

(1061) 
Let us now attempt to derive an equation of this form from Maxwell's equations.
Maxwell's equations are written:



(1062) 



(1063) 



(1064) 



(1065) 
We can cross Eq. (1065) divided by with , and rearrange, to give

(1066) 
Next, let us cross with Eq. (1064) times ,
rearrange, and add the result to the above equation. We obtain

(1067) 
Next, making use of Eqs. (1062) and (1063), we get
Now, from vector field theory,

(1069) 
with a similar equation for . Hence, Eq. (1068)
takes the form
Finally, when written in terms of components, the above equation becomes
since
,
and
.
Here, is a Kronecker delta symbol (i.e.,
if , and otherwise).
Comparing the above equation with Eq. (1061), we conclude that
the momentum flux density tensor of electromagnetic fields takes the
form

(1072) 
The momentum conservation equation (1061) is sometimes written

(1073) 
where

(1074) 
is called the Maxwell stress tensor.
Next: Electromagnetic radiation
Up: Electromagnetic energy and momentum
Previous: Electromagnetic momentum
Richard Fitzpatrick
20060202