In a Van de Graaff type generator a spherical metal shell is to be a 15 x 10⁵ V electrode. The dielectric strength of the gas surrounding the electrode is 5 x 10⁷ Vm^–1. What is the minimum radius of the spherical shell required? (You will learn from this exercise why one cannot build an electrostatic generator using a very small shell which requires a small charge to acquire a high potential.)

Solution

Given,
Dielectric strength of gas surrounding the electrodes-E = 5 x 10⁷ V / m.
Potential of spherical metal shell-V = 15 x 10⁵ V

For spherical shell,

V = \frac{1}{4 {πε}_{0}} (\frac{q}{r})

V = \frac{k q}{R}

... (i)

E = \frac{k q}{R^{2}}

... (ii)

On dividing the above equations, we get,

E = \frac{V}{R}

[ from i and ii]

R = \frac{V}{E} = 0.3 m = 30 cm R = \frac{15 \times 10^{5}}{10 % of 5 \times 10^{7}} R = \frac{15 \times 10^{5}}{\frac{10}{100} \times 5 \times 10^{5} \times 100} = \frac{3}{10} m R = 30 cm .

Minimum radius of the spherical shell required-r = 30 cm.

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