WOLFRAM|DEMONSTRATIONS PROJECT

Cylindrical Cavity Resonator

​
configuration (in m):
R
0.3
d
0.6
mode
TE
TM
m
0
1
2
3
n
1
2
3
p
1
2
3
An electromagnetic wave can be confined inside a space surrounded by conducting walls, which is called a cavity. Consider a cylindrical cavity with inner radius
R
and height
d
. There are two possible wave modes: transverse electric (TE) and transverse magnetic (TM). For appropriate field variables for those modes, separation of variables leads to harmonic solutions to the wave equation (from Maxwell's equations) of the form
J
m
(hr)
imφ
e
f(z)
, where
J
m
(x)
is a Bessel function of the first kind. The constant
m
is an integer
m=0,1,2,…
. Noting that
h
is an eigenvalue of the Helmholtz equation and taking into account the boundary condition for the trigonometric function
f(z)
, introduce the additional integer indices:
n=1,2,…
,
p=1,2,…
(TE), and
p=0,1,2,…
(TM).
The possible electromagnetic resonances can be classified as
TE
mnp
or
TM
mnp
, which completely determine the electromagnetic fields
E
and
H
in the cavity. Resonance states show localization of energy density in the cylindrical cavity. The two contributions to energy density, electric
w
e
=(
ϵ
0
/2)E
2
|
and magnetic
w
m
=(
μ
0
/2)H
2
|
, can be identified, with the total energy density given by
w=
w
e
+
w
m
.
This Demonstration shows the three-dimensional distributions of the energy densities
w
e
and
w
m
in normalized bases for the
TE
mnp
and
TM
mnp
modes within the cylinder. The distributions of
w
e
and
w
m
on the two planes
z=const
and
φ=const
are shown in red and blue, respectively. Considerable time is necessary to refresh the image, even with the image quality decreased.