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A long cylinder of radius R contains a uniformly distributed volume charge density rho. A particle of charge q moves in a circular orbit around the cylinder (outside it). What is the kinetic energy of the particle? (It is independent of the orbit radius.)

1. rho*q*R² / (4*e0)
2. rho*q*R² / (2*e0)
3. q*rho / (4*e0*R²)
4. 4*e0*R² / (q*rho)

Correct answer: rho*q*R² / (4*e0)

Solution

Outside the cylinder (distance r > R), Gauss's law gives E = lambda/(2*pi*e0*r) where lambda = rho*pi*R². So E = rho*R²/(2*e0*r). The electric force supplies centripetal force: m*v²/r = qE = q*rho*R²/(2*e0*r). Thus m*v² = q*rho*R²/(2*e0), and KE = (1/2)m*v² = q*rho*R²/(4*e0). The radius cancels, so KE is the same for any orbit.

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