##### Induced Surface Charges
ems.ca.mi.isc

Use scd_cond, with the normal direction now being \(\hat{\bf z}\):

- Gr (3.10)

\[ \sigma(x, y) = \frac{-qd}{4\pi (x^2 + y^2 + (d/2)^2)^{3/2}} \tag{scd_dip_z}\label{scd_dip_z} \]

The total induced charge can be obtained by simple integration as \(Q = \int da ~\sigma\). Using planar coordinates, \(\sigma(r) = \frac{-qd}{4\pi (r^2 + (d/2)^2)^{3/2}}\), so

- Gr (3.11)

\[ Q = \int_0^{2\pi} d\varphi \int_0^{\infty} dr r \frac{-qd}{4\pi (r^2 + (d/2)^2)^{3/2}} = \frac{qd/2}{\sqrt{r^2 + (d/2)^2}}|_0^{\infty} = -q \tag{sc_dip_z}\label{sc_dip_z} \]

This is precisely what we expected, namely that the total charge induced on the surface of the conductor at \(z=0\) precisely compensates the point charge \(q\) above it.

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Author: Jean-Sébastien Caux

Created: 2022-03-24 Thu 08:42