A circuit that takes time to respond. Charge a capacitor through a resistor and its voltage climbs the exponential curve VC = ε(1 − e−t/τ); flip the switch to discharge and it falls back the same way, ε·e−t/τ. The pace is set entirely by the time constant τ = R·C — after one τ the capacitor has covered 63% of its change, and after about 5τ it is effectively done. Set the EMF, resistance and capacitance, throw the switch, and watch the plates fill, the current dots slow, and the curve sweep — live on your device.
You are in the Physics lab.
What each moving part of the picture is telling you.
On the left is the real circuit: battery, switch, resistor and capacitor in a loop. The shading between the capacitor plates tracks VC — it darkens as the capacitor fills and fades as it empties. The blue dots are the current: they race around the loop at the instant of switching and slow down as the current decays toward zero, because I = (ε/R)e−t/τ. When the switch reads Discharge the battery greys out — it is no longer in the circuit.
The right panel plots the capacitor voltage VC (blue) and the current I (amber) against time measured in units of τ. The dashed guide marks the 63% point (charging) or 37% point (discharging) reached at exactly t = τ, and the green line at 5τ marks the ≈99% point where the capacitor is effectively fully charged or discharged. The bright dot sweeps along the voltage curve at the current instant.
Every live readout follows from τ = R·C. Drag R or C up and τ grows, so the curve stretches out and the plates fill more slowly; drag them down and it all happens in a flash. The EMF ε sets the height the voltage climbs to and the charge Q = C·VC stored at the top. Try the presets — Fast τ, Slow τ, Big capacitor and Discharge from full — to feel how each knob changes the pace.
An exponential approach to a limit, governed by a single time constant.
Throw the switch to Charge and current floods in, but as charge piles onto the capacitor its voltage rises to oppose the supply, so the current tapers off. The voltage grows along VC = ε(1 − e−t/τ) — steep at first, then flattening toward ε — while the current falls as I = (ε/R)e−t/τ. The stored charge is Q = C·VC, climbing the same curve as the voltage.
Flip to Discharge and the battery drops out of the loop. The capacitor now drives the current, so its voltage falls along VC = ε·e−t/τ — fast then slow — and the current decays with the same magnitude, (ε/R)e−t/τ, just flowing the other way. After one τ the voltage has dropped to 37% of where it started.
The rate the voltage changes is proportional to how far it still has to go: the charging current is set by the remaining gap (ε − VC)/R, the defining property of exponential approach to a limit. Solving that gives the e−t/τ form with τ = R·C. That is why the curve only ever approaches its target — 63% of the way there after one τ, and about 99% after five.
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