Coulomb Equilibrium of Charges

Surround a spot with fixed charges and you can always find a point where their pulls and pushes cancel, a place where a test charge feels no net force at all. It looks like you could balance a charge there. You cannot. Earnshaw's theorem says no arrangement of fixed charges can hold another charge in stable equilibrium: every balance point is a saddle, a hill in one direction and a valley in the other, so the charge is stable against a nudge one way but rolls off the moment it drifts the other way. The scene shows the potential landscape as contours over a colour map, marks the balance point, and lets you release a test charge to watch it slide off; drag the fixed charges to reshape the landscape. The diagnostic takes two slices of the potential through the balance point, one a valley and one a hill, the signature of a saddle and the whole reason the trap is impossible.

Figure 1. Coulomb equilibrium and Earnshaw's theorem. Top: the electric potential of fixed point charges (equipotential contours over a diverging colour map); the force-free balance point is marked, and a released test charge slides off it. Charges are draggable. Bottom: slices of the potential through the balance point, along the two in-plane axes and out of the plane (z); their curvatures sum to zero (Laplace), so there is always a hill, and no stable trap. Method: Coulomb superposition, F = sum k q q_j (r - r_j) / |r - r_j|^3.

chargestwo +

test+ test

WHAT TO TRY

Press Reset to drop the test charge right at the balance point: it sits a moment, then slides off and escapes.
Drag the test charge anywhere and let go; it never settles, it always runs to a charge or off to infinity.
Read the lower plot: through the balance point the potential is a valley one way and a hill the other, a saddle.
Drag the fixed charges to reshape the landscape; the balance point moves, but it is always a saddle.