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Answer :
Final answer:
The minimum radius required for the Van de Graaff generator's spherical metal shell to prevent dielectric breakdown is 3 meters, corresponding to option D.
Explanation:
The subject of the question is to find the minimum radius of a spherical metal shell in a Van de Graaff generator where the dielectric strength of the surrounding gas is known, and the shell's potential is given. The formula to use is V = kQ/r, where V is the potential on the sphere, Q is the charge, and r is the radius of the sphere. However, in this context, we are concerned with the dielectric strength and potential leading to a breakdown, so we're interested in the maximum electric field E that can exist at the surface of the sphere before breakdown occurs, given by E = V/r. By rearranging the formula to solve for r, we have r = V/E. Given that the potential V is 15 × 10⁶ V and the dielectric strength E of the gas is 5 × 10⁷ Vm⁻¹, the minimum radius r can be calculated.
Substituting the given values:
r = V/E = (15 × 10⁶ V) / (5 × 10⁷ Vm⁻¹) =
r = 3 meters
Therefore, the minimum radius required for the spherical metal shell to prevent dielectric breakdown is 3 meters, which corresponds to option D in the multiple-choice answers.
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