Electrostatics — Advanced Multi-Step Practice (Hard)

Quick Recap — Advanced Electrostatics (Multi-step)

• Charge sharing / redistribution: common potential $V=\dfrac{C_1V_1+C_2V_2}{C_1+C_2}$; heat lost $=\dfrac12\dfrac{C_1C_2}{C_1+C_2}(V_1-V_2)^2$.
• Coalescing drops: $n$ drops (radius $r$, potential $V$) $\to$ one drop of $V'=n^{2/3}V$ (since $R=n^{1/3}r$, $Q=nq$).
• Uniformly charged solid sphere: inside $E=\dfrac{kQr}{R^3}$, $V_{centre}=\dfrac{3kQ}{2R}=1.5\,V_{surface}$; self-energy $\dfrac{3kQ^2}{5R}$.
• Concentric shells ($+q$ at $a$, $-q$ at $b>a$): outside field $=0$; $V_a=kq\!\left(\dfrac1a-\dfrac1b\right)$.
• Slab of thickness $d/2$ (full area): $C=\dfrac{2K}{K+1}C_0$. Touching identical spheres share charge equally: new charges $\dfrac{q_1+q_2}{2}$ each.

Worked mini-example. $+q$ and $-3q$ attract with force $F=\dfrac{k\,q(3q)}{r^2}$. Touched together they share net charge $-2q$, i.e. $-q$ each; returned to distance $r$ the force is $\dfrac{k\,q^2}{r^2}=\dfrac{F}{3}$, now repulsive.