# 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