Pre-Quantum Electrodynamics

• Gr 3.2.3

The force between the point charge and the plane is the same as in the case of two point charges:

\[ {\bf F} = -\frac{1}{4\pi \varepsilon_0} \frac{q^2}{d^2} \hat{\bf z} \label{Gr(3.12)} \]

The energy is however not exactly the same: for two point charges, W_pcd gives \[ W = -\frac{1}{4\pi \varepsilon_0} \frac{q^2}{d} \] whereas for the single charge and conducting plane,

\[ W = -\frac{1}{4\pi \varepsilon_0} \frac{q^2}{2d} \] The simple reason for the \(1/2\) factor between these two results is that in the latter case, we are integrating \(E^2\) in W_intEsq but over only half the space as in the pure dipole case.