4.5 RC Circuits

Charging an RC Circuit

When a capacitor charges through a resistor, current and voltage change exponentially over time. Charge follows q(t) = Q₀(1 − e^(−t/RC)), and current is I(t) = (V/R)e^(−t/RC). The time constant τ = RC measures how quickly charging occurs — larger τ means slower charging. At t = τ, the capacitor reaches about 63% of full charge.

Discharging an RC Circuit

When a charged capacitor discharges through a resistor, charge decays as q(t) = Q₀e^(−t/RC), with current I(t) = −(Q₀/RC)e^(−t/RC). Both quantities decrease exponentially toward zero, never truly reaching it. After one time constant τ, only about 37% of the original charge remains.

Differential Equation Approach

Applying Kirchhoff's voltage law to an RC loop gives a first-order differential equation: for charging, ε = IR + q/C, and since I = dq/dt, this becomes R(dq/dt) + q/C = ε. Solving this via separation of variables yields the exponential charge/discharge functions — a hallmark FRQ skill in AP Physics C.

Key terms

Time constant (τ)

τ = RC; the time for charge/current to change by a factor of e (~63% charged or ~37% remaining).

Charging equation for q(t)

q(t) = Q₀(1 − e^(−t/RC)), where Q₀ = Cε is the maximum charge.

Discharging equation for q(t)

q(t) = Q₀e^(−t/RC), decaying exponentially from initial charge Q₀.

KVL equation for charging RC circuit

ε = IR + q/C → R(dq/dt) + q/C = ε, solved via separation of variables.

Current during discharging

I(t) = −(Q₀/RC)e^(−t/RC); current decreases exponentially and flows in the direction opposite to charging.

Try an AP-style question

What is the charge on a charging capacitor as a function of time?

1. q(t) = Cε(1 − e^(−t/RC))
2. q(t) = Cε e^(−t/RC)
3. q(t) = Cε(1 − t/RC)
4. q(t) = Cε t/RC