A uniformly charged ring of radius carrying total charge produces

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Foundations of mathematical language · Variational and asymptotic methods

Mathematical Methods for Physics, Ch Series, limits and controlled approximation

Problem

A uniformly charged ring of radius aa carrying total charge QQ produces, on its axis at distance z>0z > 0 from its centre, an exact electric field E(z)=14πϵ0Qz(z2+a2)3/2.E(z) = \frac{1}{4\pi\epsilon_0}\frac{Qz}{(z^2+a^2)^{3/2}}. ((a)) For z≫az \gg a, use the binomial series with the non-integer exponent −3/2-3/2 to expand E(z)E(z) in powers of (a/z)2(a/z)^2 up to and including that term, writing the discarded term in OO notation, and show the leading term is the point-charge field Q/(4πϵ0z2)Q/(4\pi\epsilon_0 z^2). ((b)) Evaluate the first-order fractional correction to the point-charge field at z=5az = 5a, and compare it with the exact fractional correction from the closed form.

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