Charging and discharging

Nothing crosses the gap

Before the graphs, what the charge is actually doing, because the word charging suggests something being poured into a container and that is not the process at all. An insulator sits between the plates, and no charge crosses it at any stage.

Charging makes the supply a pump. It drags electrons off the plate wired to its positive terminal and pushes the very same number onto the plate wired to its negative terminal. The first plate is left short of electrons and so carries +Q, the second holds the excess and carries −Q, and the two match exactly because one current threads the single loop, feeding one plate at the same rate it drains the other. An ammeter in either lead reads the same current at every instant, and no charge flows across the gap; what changes there is the electric field, growing as the plates charge, which is where the stored energy sits.

That flow throttles itself. Each electron added to the negative plate repels the next one on its way, and each electron removed from the positive plate makes the next one harder to remove, so the pd across the capacitor climbs and opposes the supply. The pump slows as it works, and it stops when the capacitor's pd has risen to match the supply's.

Discharging is the same traffic in reverse. Take the supply away and offer the electrons a path through a resistor from the crowded plate to the depleted one, and the pd between the plates drives them round it. The two plates neutralise each other through the external circuit, never across the gap, and the current stops when there is no excess left anywhere to drive one.

Discharge, and the shared exponential

Let a charged capacitor drive current through a resistor and three quantities fall together. The pd tracks the charge through V = Q/C, the current tracks the pd through I = V/R, so Q, V and I all follow one falling curve. The shape explains itself. A full capacitor pushes a large current and empties quickly, but the emptier it gets the smaller the pd, so the smaller the current and the slower the emptying. Each step of loss weakens the very thing driving the loss, and that self-throttling is what makes the decay exponential, steep while full, gentle while nearly empty, and never quite reaching zero.

Charging, the odd one out

Connect an uncharged capacitor to a supply through a resistor and the charge and the capacitor pd climb towards their final values, mirroring the discharge curves. The current does the opposite: at the first instant the capacitor is uncharged, offers no opposing pd, and the current starts at its maximum, I₀ = V/R. As charge builds, the capacitor's growing pd opposes the supply, the difference across the resistor shrinks, and the current falls away towards zero. Fully charged means the capacitor pd equals the supply and nothing flows. Two quantities rise and one falls, and nothing here happens at a steady rate.

Gradients and areas

You are asked to read these graphs, not only to sketch them, and the dictionary is one you already own. On the Q against t graph the gradient gives the current, since I = ΔQ/Δt, and it is steepest at the start of either process and flattens towards zero. Fix the convention before you write a number down. Every graph, equation and answer here treats the discharge current as a magnitude, a positive quantity falling towards zero. The gradient of Q against t comes out negative during a discharge, and that minus sign says only that charge is leaving the plates, so take its size and move on.

On the I against t graph the area underneath is the charge transferred, the same reading as in the electricity unit. Over a full charge, the total area under the decaying current curve equals the final charge CV, however the current wriggles on the way.