Question 2.51: Find the potential on the rim of a uniformly charged disk (r...

Introduction to Electrodynamics – Solution Manuals [EXP-2863]

Find the potential on the rim of a uniformly charged disk (radius R, charge density σ). [Hint: First show that $V=k\left(\sigma R / \pi \epsilon_{0}\right)$ , for some dimensionless number k, which you can express as an integral. Then evaluate k analytically, if you can, or by computer.]

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$V=\frac{1}{4 \pi \epsilon_{0}} \int \frac{\sigma}{ᴫ} d a=\frac{\sigma}{4 \pi \epsilon_{0}} \int_{0}^{R} \int_{0}^{2 \pi} \frac{1}{\sqrt{R^{2}+s^{2}-2 R s \cos \phi}} s d s d \phi$ .

\text { Let } u \equiv s / R \text {. Then }

$V=\frac{2 \sigma R}{4 \pi \epsilon_{0}} \int_{0}^{1}\left(\int_{0}^{\pi} \frac{u}{\sqrt{1+u^{2}-2 u \cos \phi}} d \phi\right) d u$ .

The (double) integral is a pure number; Mathematica says it is 2. So

$V=\frac{\sigma R}{\pi \epsilon_{0}}$ .

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