A positive charge Q is uniformly distributed throughout the volume of a nonconducting sphere of radius R. A point mass having charge +q and mass m is fired towards the centre of the sphere with velocity v from a point at distance r(r > R) from the centre of the sphere. Find the minimum velocity v so that it can penetrate R/2 distance of the sphere. Neglect any resistance other than electric interaction. Charge on the small mass remains constant throughout the motion.
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Question

A positive charge Q is uniformly distributed throughout the volume of a nonconducting sphere of radius R. A point mass having charge +q and mass m is fired towards the centre of the sphere with velocity v from a point at distance r(r > R) from the centre of the sphere. Find the minimum velocity v so that it can penetrate R/2 distance of the sphere. Neglect any resistance other than electric interaction. Charge on the small mass remains constant throughout the motion.

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G1 Physics · Accepted Answer

Step 1: In electrostatic equilibrium, E = 0 inside a conductor and the conductor is an equipotential. Surface charges adjust so the boundary condition field just outside·𝐧 = σ/ε₀ is satisfied. Step 2: For connected conductors set their potentials equal; for sheets use electrostatic pressure p = ε₀E²/2 when appropriate. Step 3: Apply charge conservation and the given geometry. Final: A: √[(2KQq/(mR))((r−R)/r + 3/8)].