Statement
A spherical conducting shell of radius a centered at the origin has a potential field which is constant inside the sphere and has an inverse-r dependence outside the sphere. The zero reference is at infinity. Find an expression for the stored energy that this potential represents.
System Parameters
Permittivity of free space:
Inside/on the shell:
Outside the shell:
Solution
First calculate the electric field:
Inside/on the shell:
Outside the shell:
Writing this explicitly,
P.16 |
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P.17 The Spherical Capacitor |
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The energy is then given by the volume integral over the electric field squared.
Solving this symbolically,
Examine the correspondence between the size of the sphere and the energy stored in the electric field. Also, note that the total charge on the shell, by Gauss' law, is
The energy is equal to 1/2*QV, the familiar result for the energy stored in a capacitor (in this case, a spherical capacitor of infinite radius). Try to get a feeling for what capacity different size spheres have for energy storage, and what the corresponding voltages and field strengths are. Suppose you wanted to store a microjoule of energy with a 1 volt potential at the sphere's surface. How big a sphere is required? Now suppose there are 1000 volts at the surface. How big is the sphere?
