A common method for solving Poisson's equation in electrostatics is to patch together two or more solutions of Laplace's equation using boundary conditions on the potential and its gradient. Other methods may generate solutions without the need to check these conditions explicitly, and reconciling these solutions with the appropriate boundary conditions can be surprisingly subtle. As a result, a student may arrive at paradoxical conclusions—even in the case of elementary problems—that seem to be at odds with basic physical intuition. We illustrate this issue by showing how the potential of a uniformly charged ring appears to violate continuity of the normal component of the electric field at a chargeless surface.

Topics
Electromagnetism, Electric fields, Electrostatics, Maxwell equations, Partial differential equations, Coordinate system, Order theory, Generalized functions, Educational assessment
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For instance, Stirling's approximation ℓ!∼ℓ ℓ e−ℓ 2π ℓ can be used to verify that the summand grows as ℓ for large ℓ, and therefore does not vanish in the ℓ → ∞ limit. Combining subsequent even (ℓ = 2k) and odd (ℓ = 2k + 1) terms in the sum gives a strictly negative summand proportional to 1/k at large k. This form of the sum can be shown to diverge by the integral test or by comparison with the harmonic series.
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Students will have encountered a similar Legendre series in problem 3.22 of Ref. 2.

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One must use identities for Legendre polynomials on the half-intervals 0 < θ < π/2 and π/2 < θ < π, where the familiar orthogonality conditions do not apply.
© 2014 American Association of Physics Teachers.
2014
American Association of Physics Teachers
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