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Two non-conducting hemispherical shells, each carrying a uniform surface charge density sigma, rest on a smooth horizontal floor and are held by ideal springs so that the two hemispheres just touch to form a full sphere. Determine the compression in each spring needed to balance the electrostatic repulsion between the two hemispheres.

1. sigma² * R² / (2 * epsilon0 * k)
2. sigma² * pi * R² / (2 * epsilon0 * k)
3. sigma² * pi * R² / (epsilon0 * k)
4. None of these

Correct answer: sigma² * pi * R² / (2 * epsilon0 * k)

Solution

A charged surface experiences an outward electrostatic pressure P = sigma²/(2*epsilon0). The net force tending to separate two hemispheres of a charged sphere equals this pressure times the equatorial disc area pi*R²: F = (sigma²/(2*epsilon0))*pi*R². In equilibrium each spring supplies this force: k*x = sigma²*pi*R²/(2*epsilon0), giving x = sigma²*pi*R²/(2*epsilon0*k).

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