The capacitor: storing charge and energy
So far current has flowed freely through resistors. A capacitor is different: it is two conductors separated by a gap, and it stores charge by piling positive on one plate and negative on the other. The more charge you pack on for each volt across it, the larger its capacitance.
Charge stored equals capacitance times the voltage across the plates. Capacitance is measured in farads: 1 F = 1 coulomb per volt.
Charging through a resistor: the RC circuit
Connect a battery, a resistor and an initially empty capacitor in series and close the switch. The capacitor does not fill instantly — the resistor throttles the current. At first, with the capacitor empty, current is largest; as charge builds and the capacitor's voltage climbs to oppose the battery, the current tapers off. This is the RC circuit, and both charge and current follow smooth exponential curves.
While charging, the charge rises from zero and approaches its full value Cε along an exponential curve.
The current starts at its maximum ε/R and decays to zero as the capacitor fills.
The time constant sets the clock
Every exponential in the RC circuit is governed by one number, the time constant \tau = RC. It has units of seconds and sets the pace: after one time constant the capacitor has reached about 63% of full charge; after roughly five time constants it is over 99% charged — effectively full.
The time constant is resistance times capacitance; bigger R or C means slower charging and discharging.
Discharging: a charged capacitor releases its charge, decaying exponentially with the same time constant τ = RC.
Putting the whole track together
Look how far you have come. You can now say precisely what current is, compute resistance from a material and its shape, apply Ohm's law and three forms of power, reduce series-and-parallel networks, account for a real battery's internal resistance, solve any tangled circuit with Kirchhoff's two rules, and track a capacitor charging on its exponential clock. That is the working toolkit of DC circuit analysis — the same one behind every gadget you own.
Where does this lead? Everything here was steady direct current. The moment currents start to change, a new world opens: a moving charge or a current creates a magnetic field (the Magnetism track), a changing magnetic field pushes a current back (the Induction track), and a capacitor paired with a coil trades energy back and forth to oscillate, giving us alternating current and, ultimately, the electromagnetic waves that are light itself. The humble battery-and-bulb loop was the first step onto that road.