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