Question

A metal sphere of radius R is charged to a potential V. (a) Find the electrostatic energy stored in the electric field within a concentric sphere of radius 2 R. (b) Show that the electrostatic field energy stored outside the sphere of radius 2 R equals that stored within it.

Answer 1

(a) Electro static energy UE = π∈0 RV2

(b) The electrostatic field energy stored outside the sphere of radius 2 R equals that stored within it.

Explanation:

The energy density a distance r from the centre of the sphere is given by  

$U_{E}=\frac{1}{2} \varepsilon_{0} E^{2}=\frac{1}{2} \varepsilon_{0}\left(\frac{q}{4 \pi \varepsilon_{0} r^{2}}\right)^{2}=\frac{q^{2}}{32 \pi^{2} \varepsilon_{0} r^{4}}$

Now consider a spherical element of radius r and thickness dr around the sphere f radius R (r>R).

The volume of sphere element is given by  dV = 4πr2dr

Energy stored in the element is given by  

$\mathrm{dU}_{\mathrm{E}}=\mathrm{u}_{\mathrm{E}} \times \mathrm{dV}=\frac{q^{2}}{32 \pi^{2} \varepsilon_{0} r^{4}} \times 4 \pi \mathrm{r}^{2} \mathrm{dr}=\frac{q^{2}}{8 \pi \varepsilon_{0}} \frac{d r}{r^{2}}$

Since q = 4π€RV

Therefore, U = π€RV2

Thus, the electrostatic energy stored outside of the sphere f radius 2R equals that stored with in it.