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A potential function satisfying Laplace's equation in a region can have a maximum

  • A.anywhere within the region
  • B.only at the centre of the region
  • C.nowhere at all
  • D.only on the boundary of the region

Answer: D

The mean value property of a harmonic function makes the potential at any interior point the average over a small sphere about it, so no interior point can exceed all its neighbours. This is why a charge cannot be held in stable electrostatic equilibrium by fixed charges alone, which is Earnshaw's theorem.

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