= Charge sharing between touching conducting spheres
For externally tangent <conducting spheres> of radii $a,b$, centre separation $D=a+b$, and common <electric potential> $V$, the <method of images> gives
$$
Q_a=4\pi\epsilon_0Vab\sum_{n=0}^{\infty}\left(\frac1{b+nD}-\frac1{(n+1)D}\right),\quad
Q_b=4\pi\epsilon_0Vab\sum_{n=0}^{\infty}\left(\frac1{a+nD}-\frac1{(n+1)D}\right).
$$
Alternating spherical image reflections cancel the nonconstant boundary <electric potential>. Pairing terms makes each series convergent. The ratio is generally not $a:b$: that rule uses isolated <capacitances> and applies to widely separated <conducting spheres> connected by a wire. For $b=2a$, the smaller sphere carries the fraction $1/2-\pi/(6\sqrt3\ln3)\simeq0.22483$ of the total <electric charge>.
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