The energy density U of a continuous charge distribution, according to
popular belief, is
U = (1/2)*E^2
where E is the electric field strength and all fundamental constants
equal one. I intend to show that this should, instead, be
U = E^2
In order to simplify, I'll start with the energy of a point charge
distribution. The energy of the distribution is just the work required
to assemble the distribution. Let's start with a pair of identical
charged particles, e1 and e2, one at +infinity and the other at
-infinity. The work W12 required to bring e1 in from infinity to zero
against the field of e2 is
W12 = e1(e2/r12) (1)
where e2/r12 is the potential due to e2 and r12 is the distance between
e1 and e2 after we're through bringing e1 in. But since e2 is not in the
picture, yet (r12=infinity), the work required to bring e1 in is zero,
i.e., W12 = 0.
The work W21 necessary to bring in e2 is
W21 = e2(e1/r21)
BUT, as we bring e2 in, we must do work on e1 against the field of e2 to
keep it in place at the origin. The work necessary to keep e1 where it
is, is the same as the work required to bring e1 in against the field of
e2, had r12 been equal to r21. But this is the same as (1) for r12 =
r21. So the total work W required to assemble the two particles is
W = W12 + W21 = e1(e2/r12) + e2(e1/r21) (2)
or
W = sum(i = 1,2)sum(j = 1,2){ei(ej/rij)}
for i =/= j.
where "=/=" means "not equal to".
For three particles, we have to add the work required to bring in e3
against the fields of both e1 and e2
W31 + W32 = e3(e1/r31) + e3(e2/r32) (3)
But again, we have to include the work required to keep e1 and e2 in
position as we bring in e3. The work W13 and W23 required to keep e1 and
e2 in position is
W13 = e1(e3/r13) (4)
and
W23 = e2(e3/r23) (5)
respectively. So the total work W required to assemble the three
particles, from (2), (3), (4) and (5), is
for i =/= j. So for any number of particles n, we can apparently write
W = sum(i = 1 to n)sum(j = 1 to n){ei(ej/rij)}
for i =/= j. We can also write this as
W = sum(i = 1 to n){ei}[sum(j = 1 to n){ej/rij}]
for i =/= j. The term in brackets is the potential V(ri) at point ri
(the position of ei) due to all other charges. Thus,
W = sum(i = 1 to n){ei*V(ri)} (6)
For a volume charge density rho, (6) becomes
W = \int{rho*V dv}
where "\int" means integral, V is the potential, and dv is the volume
element. This can also be written as
W = \int{E^2 dv}
where E is the electric field strength. This represents the energy
stored in the configuration, therefore, we can interpret E^2 as the
energy density U
U = E^2 (7)
However, physics texts write the energy density as
U = (1/2)*E^2 (8)
Clearly the texts are incorrect. The correct description of the energy
density, as I have shown, should be (7) not (8).