Energy Stored in an Electric Field — Question 6

Question 6

The exterior energy of a charged sphere has a long radial tail

An isolated conducting sphere of radius R carries charge Q in vacuum. Its exterior field is $E=Q/(4\pi\epsilon_0r^2)$ and its interior field is zero.

1. Integrate the field energy from R to infinity.

2. Find the fraction of energy outside radius nR, n>1, and the radius containing 90 percent of the field energy.

Question 6 - Solution

The exterior energy of a charged sphere has a long radial tail

(a) Method and calculation

$$U=\int_R^\infty\frac{\epsilon_0E^2}{2}4\pi r^2dr=\frac{Q^2}{8\pi\epsilon_0}\int_R^\infty\frac{dr}{r^2}.$$$$U=\frac{Q^2}{8\pi\epsilon_0R}.$$

(b) Interpretation and further calculation

$$U_{r>nR}/U=1/n.$$$$U_{R<r<r_{90}}/U=1-R/r_{90}=.900\Rightarrow r_{90}=10R.$$

Physical check

The energy density falls as inverse fourth power, but the growing spherical shell volume makes the energy tail fall only as inverse radius.