Pre-Quantum Electrodynamics

Use (\ref{Gr(2.49)}), where normal direction is now \(\hat{\bf z}\). \[ \sigma(x, y) = \frac{-qd}{2\pi (x^2 + y^2 + d^2)^{3/2}} \label{Gr(3.10)} \] Total induced charge: \(Q = \int \sigma da\). Use planar coordinates: \(\sigma(r) = \frac{-qd}{2\pi (r^2 + d^2)^{3/2}}\), so \[ Q = \int_0^{2\pi} d\phi \int_0^{\infty} dr r \frac{-qd}{2\pi (r^2 + d^2)^{3/2}} = \frac{qd}{\sqrt{r^2 + d^2}}|_0^{\infty} = -q \label{Gr(3.11)} \]