WOLFRAM|DEMONSTRATIONS PROJECT

Electromagnetic Field Energies in Capacitors and Inductors

A capacitor with square plates of width
a
separated by a distance
d
with a filler of dielectric constant (relative permittivity)
κ
has a capacitance given by
C=κ
ϵ
0
2
a
/d
. Typical values are in the range of picofarads (pF). A voltage
V
can hold positive and negative charges
q=±CV
on the plates of the capacitor while producing an internal electric field
E=V/d
. Assuming idealized geometry, the energy of a charged capacitor equals
1
2
C
2
V
. This energy can be considered to be stored in the electric field, which implies a corresponding energy density
ρ
elec
=
1
2
ϵ
2
E
(with
ϵ=κ
ϵ
0
).
Next consider an air-core inductor, again assuming idealized geometry. The relative permeability
κ
m
is approximated as 1. The inductance of a helical conducting coil, as shown in the graphic, is then given by
L=
μ
0
2
n
π
2
r
/ℓ
, where
n
is the number of turns. Typical values can be in the range of microhenries (
μ
H). Considered as a solenoid, the inductor produces a magnetic field
B=
μ
0
nI/ℓ
, when carrying a current
I=V/R
. The energy of the inductor equals
1
2
L
2
I
, which implies a magnetic-field energy density
ρ
mag
=
1
2
μ
0
2
B
.
Combining the above results gives the well-known formula for the energy density of an electromagnetic field in a vacuum:
ρ
em
=
1
2

ϵ
0
2
E
+
-1
μ
0
2
B

. This is valid for electric and magnetic fields from any sources, notably for electromagnetic radiation.