3.4 Electric fields and potentials of point charges, 2

A First Course in String Theory

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(b) Show that with d spatial dimensions, the potential \Phi due to a point charge q is given by

\displaystyle{\Phi(r) = \frac{\Gamma \left( \frac{d}{2} - 1 \right)}{4 \pi^{d/2}} \frac{q}{r^{d-2}}}

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Eq. (3.74):

\displaystyle{E(r) = \frac{\Gamma\left( \frac{d}{2} \right)}{2 \pi^{\frac{d}{2}}} \frac{q}{r^{d-1}}}

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\displaystyle{\begin{aligned}        E(r) &= -\frac{d\Phi(r)}{dr}  \\ \\    \int_r^{\infty} d\Phi &= - \frac{\Gamma\left( \frac{d}{2} \right)}{2 \pi^{\frac{d}{2}}} q \int_r^{\infty} r^{-d+1} dr  \\ \\    \Phi(\infty) - \Phi(r) &= \frac{\Gamma\left( \frac{d}{2} \right)}{4 \pi^{\frac{d}{2}}} \frac{q}{\frac{d}{2}-1} \left[\frac{1}{r^{d-2}} \right]_r^{\infty}   \\ \\    \Gamma (z) &= \frac{\Gamma (z+1)}{z} \\ \\     \Phi(r) &= \frac{\Gamma\left( \frac{d}{2} - 1\right)}{4 \pi^{\frac{d}{2}}} \frac{q}{r^{d-2}}    \\ \\         \end{aligned}  }

— Me@2023-05-17 09:11:07 AM

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2023.05.17 Wednesday (c) All rights reserved by ACHK