Past exam of the mathematics course of the University of Cambridge / 2014 / ib / Paper 1 / 16A (Electromagnetism)

An ideal conductor in electrostatic equilibrium has zero field in its interior and constant potential on its surface. Grounding fixes that potential to zero. The tangential field is continuous and therefore vanishes; the normal field has the jump fixed by the induced surface charge density $\sigma$:Here the normal points upward out of the conductor. The normal field is not generally zero; grounding prescribes the potential, not the induced charge density.

The method of images places an image charge $-q$ at the reflected position below the plane. The candidate potential in the physical upper half-space isAt $z=0$ the two terms cancel, so $\Phi =0$ and $E_x=E_y=0$. Direct differentiation givesThus the upper-side field satisfies exactly the conductor boundary conditions from part (i). The fictitious pair does have a field below the plane, but that part of its solution is not used for the physical conductor.

In $z>0$, the potential has the correct real point source, the correct zero Dirichlet boundary condition, and the required decay at infinity. By Uniqueness of the Dirichlet problem, it is the physical potential there. This is why the artificial charge can represent the induced conductor response without being a second physical charge.

As the real charge descends to height $z(t)$, its instantaneous image charge is at $-z(t)$. The force is thereforeThe image is not kept at its initial position. Multiplying by $\dot{z}$ and using $z(0)=d$, $\dot{z}(0)=0$ givesFor the falling charge above a grounded conducting plane, the arrival time isPut $z=d{\sin }^2\theta$; the integral becomes $2d^{3/2}{\int }_0^{\pi /2}{\sin }^2\theta d\theta =\pi d^{3/2}/2$. HenceThis is the finite arrival time in the specified ideal force model, for $q\ne 0$. The field-force derivation also avoids incorrectly using the full real-image pair interaction energy: the conductor's effective energy is $-q^2/(16\pi {\epsilon }_{0}z)$, with the factor one-half appropriate to induced charge.