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A uniformly charged solid non-conducting sphere has a total positive charge Q and radius R . A narrow, smooth radial tunnel is drilled to its centre. A particle of mass m and positive charge q is projected from the surface of the sphere into the tunnel. The minimum initial speed required for the particle to reach the centre of the sphere is:

Options

  1. AqQ 2 ₀ mR
  2. B0
  3. CqQ 8 ₀ mR
  4. DqQ 4 ₀ mR

Correct answer

D. qQ 4 ₀ mR

Step-by-step solution

The electrostatic potential at the surface of the solid sphere is: V_s = 1 4 ₀ Q R The electrostatic potential at the centre of the solid sphere is: V_c = 1 4 ₀ 3Q 2R The change in potential energy of the particle as it moves from the surface to the centre is: U = q(V_c - V_s) = q ( 3Q 8 ₀ R - Q 4 ₀ R ) = qQ 8 ₀ R By the conservation of mechanical energy, the minimum initial kinetic energy must be equal to the increase in potential energy: 1 2 mv^2 = U 1 2 mv^2 = qQ 8 ₀ R v^2 = qQ 4 ₀ mR v = qQ 4 ₀ mR Answer: qQ 4

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