Quick Calculation 3.6

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A First Course in String Theory

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Verify that for \displaystyle{d=3} equation (3.74) coincides with (3.67).

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Eq. (3.67):

\displaystyle{E(r) = \frac{q}{4 \pi r^2}}

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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When \displaystyle{d=3},

\displaystyle{\begin{aligned} E(r) &= \frac{\Gamma\left( \frac{3}{2} \right)}{2 \pi^{\frac{3}{2}}} \frac{q}{r^{2}} \\ \\ &= \frac{\sqrt{\pi}}{2} \frac{1}{2 \pi \pi^{\frac{1}{2}}} \frac{q}{r^{2}} \\ \\ &= \frac{1}{4 \pi} \frac{q}{r^{2}} \\ \\ \end{aligned} }

— Me@2022-05-30 01:15:00 PM

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