A thin disc of radius carries a surface charge density ( )= k

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Vector calculus and field geometry · Vector calculus and field geometry

Conditions of Validity, Ch A coordinate system is a map: the metric, the scale factors and where a chart fails

Problem

A thin disc of radius a=0.20 ma = 0.20\ \mathrm{m} carries a surface charge density σ(ρ)=kρ,k=3.0×10−6 C m−1,\sigma(\rho)=\frac{k}{\rho}, \qquad k = 3.0\times10^{-6}\ \mathrm{C\,m^{-1}}, where ρ\rho is the plane polar radius measured from the centre of the disc. The density diverges at ρ=0\rho=0, and ρ=0\rho=0 is also the point at which the polar chart fails: det⁡g=ρ2=0\det g = \rho^2 = 0 there and the angle ϕ\phi is undefined. ((a)) Write the area element dA\mathrm{d}A of the plane polar chart in terms of det⁡g\sqrt{\det g}. ((b)) Excise a small disc of radius ε\varepsilon (0<ε<a0<\varepsilon<a) about the origin, as an improper integral is treated by excising a bad point and taking a limit, and find the total charge Q(ε)Q(\varepsilon) on the remaining annulus ε≤ρ≤a\varepsilon\le\rho\le a. ((c)) Let ε→0+\varepsilon\to0^+ and state the total charge on the disc. Explain why the coordinate failure at ρ=0\rho=0 does not prevent this limit existing.

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