In a Van de Graaff type generator a spherical metal shell is to be a $15 \times 10^{6} \mathrm{~V}$ electrode. The dielectric strength of the gas surrounding the electrode is $5 \times 10^{7}$ $\mathrm{Vm}^{-1}$. 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.)
Potential difference, $V=15 \times 10^{6} \mathrm{~V}$
Dielectric strength of the surrounding gas $=5 \times 10^{7} \mathrm{~V} / \mathrm{m}$
Electric field intensity, E = Dielectric strength = 5 × 107 V/m
Minimum radius of the spherical shell required for the purpose is given by,
$r=\frac{V}{E}$
$=\frac{15 \times 10^{6}}{5 \times 10^{7}}=0.3 \mathrm{~m}=30 \mathrm{~cm}$
Hence, the minimum radius of the spherical shell required is 30 cm.
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