Potential of a Charged Spheroid
Potential of a Charged Spheroid
This Demonstration shows the electrostatic potential of a uniformly charged spheroid. We consider both prolate spheroids, with , and oblate spheroids, with . Here , , are the semi-axes, with the axis oriented horizontally. The potential is cylindrically symmetrical and it suffices to show just the plane containing the axis. The potential external to the spheroid is given by , the sum representing a multipole expansion over the charge distribution. For an oblate or prolate spheroid, the monopole contribution is dominant, with only the quadrupole term making a significant additional contribution to the potential. The quadrupole moment of a charged spheroid is given by .
a=b<c
a=b>c
a
b
c
c
c
Φ(r)=r'=+Q+…
q
4π
ϵ
0
1
|r-r'|
3
d
1
4π
ϵ
0
q
r
1
4
3-
2
z
2
r
5
r
Q==q(-)
Q
zz
2
5
2
a
2
c
You can select the semi-axes and to display a scaled contour plot of the potential. Multiply by to find the actual potential. The same result pertains to a gravitational potential, with as the scaling factor. You can isolate the quadrupole contribution with the checkbox.
a
c
q/4π
ϵ
0
GM