TWO CHARGES
Two charges: the field vanishes at x = r/(1+√(q₂/q₁)) — but ∇²V=0 makes it a saddle, so nothing rests there. Earnshaw, drawn.
Use the simulation above to change the variables and play through the guided stages. The explanation below describes the default starting values; the simulation updates its explanation as you experiment.
Setup
Two source sites lie on a line, while q₂ may be positive, negative, or zero. The module first determines whether any finite field-null point exists and where; only then does it ask whether that stationary point could trap a test charge.
Inverse square
Like-charge fields oppose between the sources and produce one internal null.
The null
The magnitude equation gives a plus denominator for like charges and a minus denominator for unlike charges. The latter places the null outside on the weaker-magnitude charge’s side.
The saddle
At a finite field null, ∇V = 0 but ∇²V = 0 in empty space, so the stationary point must have restoring and escaping directions: a saddle, never a three-dimensional trap.
Audit
So the point of no force is a saddle, never a bowl — no static arrangement of charges can hold another still. That is Earnshaw’s theorem, and it is why matter needs quantum mechanics: Coulomb’s law alone cannot keep an atom from collapsing.