Dipole Embedded in a Linear Dielectric Sphere

​
medium
vacuum
air (dry)
silicon
water
ice
An electric dipole moment
p
is placed in the center of a linear dielectric sphere of specified medium. The resulting potential and electric field inside and outside the sphere are plotted.

Details

This Demonstration considers an electric dipole with dipole moment
p
placed inside a sphere of radius
R
composed of a linear dielectric material with dielectric constant
ϵ
r
.
Since there is no free charge, we can apply Gauss's law for the electric potential
V
, which yields the Laplace equation
2
∇
V(r,θ,ϕ)=0
.
Imposing the appropriate boundary conditions (continuity of
V
and discontinuity of
∂V
∂n
), it can be shown that the electric potentials inside and outside the sphere are given by
V
in
=
pcosθ
4πϵ
2
r
1+2
3
r
3
R
(
ϵ
r
-1)
(
ϵ
r
+2)
,r<R
,
V
out
=
pcosθ
4π
ϵ
0
2
r
3
ϵ
r
+2
,r>R
.
The corresponding electric field is given by
E=-∇V
. We plot this quantity to see the effect of different materials for the dielectric sphere. We consider the following media:
vacuum:
ϵ
r
=1
air:
ϵ
r
=1.0005
silicon:
ϵ
r
=11.7
water:
ϵ
r
=80.1
ice:
ϵ
r
=104
.

References

[1] D. J. Griffiths, Introduction to Electrodynamics, Boston: Pearson, 2013.

External Links

Wolfram Demonstration: Electrostatics

Permanent Citation

Hugo Alexis Suárez Rangel
​
​"Dipole Embedded in a Linear Dielectric Sphere"​
​http://demonstrations.wolfram.com/DipoleEmbeddedInALinearDielectricSphere/​
​Wolfram Demonstrations Project​
​Published: October 25, 2023