If a piece of metal has a charge + 0.1 μC and is placed inside a hollow metal sphere of radius 20 cm (without touching it), what is the potential of the sphere? What will the potential of the sphere become, if
(a) the sphere is temporarily earthed and then left insulated,
(b) the metal subsequently touched the inside of the sphere?

Solution

When, a piece of metal is placed inside a hollow metal sphere, an equal amount of negative charge is induced in the sphere.
Hence, surface charges of -0.1 μC and +0.1 μC are induced respectively on the inner and outer spheres.
Radius of the sphere, r = 20 cm = 0.2 m

The potential of the sphere relative to earth is determined solely by the outer surface charge q.

∴

V = \frac{1}{4 {πε}_{0}} \frac{q}{r}

Therefore,

V = 9 \times 10^{9} {Nm}^{2} C^{- 2} . \frac{0.1 \times 10^{- 6} C}{0.2 m} = 4500 V .

(i)When the sphere is temporarily earthed and then left insulated, the potential of the sphere is momentarily reduced to zero. The positive charge on the outer surface disappears but the induced negative charge inside remains. Therefore, the potential of the sphere is zero.
(ii) The induced negative charge is neutralized when the metal touches the sphere and no charge remains on the metal or sphere. Both are at the same potential that is, the sphere has zero potential.

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