Closed surface and charge balance

If $Q(t)$ is the charge on one plate of the capacitor and $i(t)$ the current in the circuit, when is $dQ/dt = +i$ and when is $dQ/dt = -i$? It seems a detail, but getting it wrong flips the sign of an entire differential equation. The answer goes through the continuity equation for charge: charge is neither created nor destroyed, it can only enter or leave through the wires.

Imagine wrapping one plate of the capacitor with an imaginary closed surface $\mathcal{S}$ (a mathematical bubble, not a physical object). The surface cuts the wire that carries current to that plate. Let $Q$ be the charge contained inside $\mathcal{S}$: it coincides with the charge on the enclosed plate, because outside the plate, in the conducting wires, there is no charge accumulation (the current simply passes through, like water in a pipe).

Continuity equation for charge

$\ev{\frac{dQ_\text{dentro}}{dt} = i_\text{entrante}}$

In words: the rate at which the charge inside $\mathcal{S}$ changes equals the current entering through the surface. It is the “charge” version of a hydraulic balance: the volume of water in a tank increases at the rate of the incoming flow and decreases at the rate of the outgoing flow.

The sign in front of $i$ in the circuit equation therefore depends on two choices we make:

• which plate is enclosed by the closed surface (positive or negative);
• the conventional direction of the current $i(t)$ in the circuit.

Collegamenti

Argomenti: Electromagnetic induction Concetti: LC and RLC oscillations · Electric current

Esercizi collegati: Problem — Natural frequency of an LC circuit · Problem — Capacitor of an AM radio receiver · Worked exercise — Forced RLC, find the generator’s EMF