Charging a Capacitor Through a Resistor

Connect an uncharged capacitor to a battery through a resistor and current starts flowing immediately. At that instant the capacitor has no charge, so it has no voltage of its own to push back — the full EMF lands across the resistor, and current is as large as it will ever get. As charge builds on the plates, the capacitor develops its own voltage, which opposes the battery. The loop equation EMF = IR + q/C shows the resistor's share shrinking as the capacitor's share grows. Less voltage across R means less current, by Ohm's law. That falling current is what makes the charging slow down over time: the circuit throttles itself.

Solving that loop equation gives q(t) = CV(1 - e^−t/RC), where RC is the time constant, the time scale that sets how fast the exponential settles. Early on, t is small compared to RC and charge rises almost linearly. As t grows past a few time constants, e^−t/RC shrinks toward zero and q creeps asymptotically toward its ceiling, CV — but mathematically it never quite gets there for any finite t. That's the same shape as radioactive decay or cooling, just climbing toward a ceiling instead of falling toward zero.

This sits right next to capacitor discharging, where q(t) = Q0 e^−t/RC with no battery to fight against, and the two share the same RC time constant. Current during charging is I(t) = (EMF/R) e^−t/RC, starting at EMF/R and decaying — the mirror image of the rising charge curve, and the natural bridge into analyzing RC circuits with calculus.