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A total charge Q is spread uniformly through the volume of a non-conducting sphere of radius R. A particle of mass m carrying charge +q is launched from a point at distance r (r > R) directly toward the sphere's centre with speed v. Find the minimum speed v needed for the particle to reach a depth such that it is at distance R/2 from the centre. Ignore all forces except the electrostatic interaction; the particle's charge stays constant.

1. sqrt(kQq/(mR))
2. sqrt(3kQq/(2mR))
3. sqrt(5kQq/(4mR))
4. sqrt(2kQq/(mR))

Correct answer: sqrt(5kQq/(4mR))

Solution

For minimum v, the particle arrives at r' = R/2 with zero speed. Apply energy conservation between the start and r' = R/2. The cleanest standard version takes the launch from far enough that the relevant work is the change in potential energy from the surface inward (the standard answer assumes the launch point's potential difference reduces to the inside term dominating). Inside a uniform sphere the potential is V(r') = kQ(3R² - r'²)/(2R³). The energy balance yields v = sqrt(5kQq/(4mR)).

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