RC Circuit in AP Physics C: Electricity and Magnetism

In AP Physics C: E&M, an RC circuit is a resistor and capacitor connected so the capacitor charges or discharges exponentially over time. The rate is set by the time constant τ = RC, and you model it with a first-order differential equation built from Kirchhoff's loop rule.

What is RC Circuit?

An RC circuit is any circuit with a resistor (R) and a capacitor (C) where the capacitor's charge changes over time instead of instantly. Hook an uncharged capacitor to a battery through a resistor and charge starts piling up on the plates. As it does, the capacitor pushes back with a growing voltage, so the current shrinks. The result is exponential behavior. Charge rises toward a max of Cε, and current dies off toward zero.

In Physics C you don't just memorize the curves. You derive them. Walking around the loop for a charging circuit gives ε − IR − q/C = 0. Since I = dq/dt, that becomes RC(dq/dt) + q = Cε. Solving it gives q(t) = Cε(1 − e^(−t/RC)) and I(t) = (ε/R)e^(−t/RC). For discharging, q(t) = Q₀e^(−t/RC). The product RC is the time constant (τ). Think of it as the circuit's built-in clock. After one τ, a charging capacitor sits at about 63% of its final charge, and a discharging one has dropped to about 37%.

Why RC Circuit matters in AP® Physics C: E&M

RC circuits are the core of Topic 11.8 in Unit 11: Electric Circuits, and they are where E&M gets genuinely calculus-based. This is the topic that turns a circuit problem into a differential equation you set up, solve, and interpret. It pulls together ideas from across the course. Capacitance and stored energy come from Unit 10, potential difference from Unit 9, and Kirchhoff's rules and conservation of charge from earlier in Unit 11. It also sets up the LR circuits in Unit 13, which follow the exact same math with an inductor swapped in. Because one setup can test derivation, graphing, limiting cases, and energy all at once, RC circuits are a favorite for multi-part free-response questions.

How RC Circuit connects across the course

Time Constant (τ) (Unit 11)

τ = RC is the single number that tells you how fast an RC circuit responds. Bigger resistance means charge trickles in more slowly. Bigger capacitance means there's more charge to fill. Either one makes the process take longer, so τ grows with both.

Capacitor and Stored Energy (Unit 10)

The capacitor's U = ½CV² and Q = CV from Unit 10 are what you plug in at every instant of an RC circuit. While charging, the battery supplies Cε² of energy. Half ends up stored in the capacitor and the other half is burned off as heat in the resistor, no matter what R is.

Kirchhoff's Loop Rule and Conservation of Charge (Unit 11)

The RC differential equation is just the loop rule with the capacitor voltage written as q/C. Charge flowing through the resistor is the same charge landing on the plates, which is why I = dq/dt links the two terms.

LR Circuits (Unit 13)

Swap the capacitor for an inductor and you get the same first-order equation with τ = L/R. Learning RC first means LR circuits feel like a rerun with the roles flipped, since an inductor fights changes in current the way a capacitor resists sudden changes in voltage.

Is RC Circuit on the AP® Physics C: E&M exam?

Expect RC circuits in both multiple choice and free response. Common moves:

• Find initial and final behavior. At t = 0 an uncharged capacitor acts like a plain wire, so the current is ε/R. One practice question asks for dq/dt at t = 0 with R = 200 Ω and ε = 12 V. That's just 12/200 = 0.06 A. After a long time (steady state), the capacitor acts like a break in the wire and the current is zero.
• Rewrite or derive the differential equation. You might be handed RC(dq/dt) + q = Cε and asked to rewrite it in terms of capacitor voltage. Substituting q = CV gives RC(dV/dt) + V = ε.
• Use the 63% / 86.5% benchmarks. One time constant gets you to 63% of max charge, two get you to 86.5%. A 2.0 μF capacitor with a 3.0 MΩ resistor has τ = 6.0 s, so reaching 63% at 6.0 s is exactly what theory predicts. If τ is 25 ms and you double R, τ becomes 50 ms, so 86.5% takes 2τ = 100 ms.
• Sketch and read graphs of q, I, or V versus time, and use a slope or a ln plot to pull out τ from data.

No released FRQ has used the phrase "RC circuit" verbatim, but the skills here (setting up a differential equation from a physical system, solving it, and checking the limits) are exactly what Physics C free-response questions reward.

RC Circuit vs LR Circuit

Both circuits follow the same exponential math, but the components behave oppositely at the extremes. In an RC circuit, an uncharged capacitor acts like a wire at t = 0 (max current) and like an open switch at steady state (zero current). In an LR circuit, the inductor acts like an open switch at t = 0 (zero current) and like a wire at steady state (max current). The time constants differ too. RC circuits use τ = RC, while LR circuits use τ = L/R.

Key things to remember about RC Circuit

• An RC circuit's charge and current change exponentially, with the time constant τ = RC setting how fast it happens.

• When charging, q(t) = Cε(1 − e^(−t/RC)) and I(t) = (ε/R)e^(−t/RC), so charge rises while current decays.

• An uncharged capacitor acts like a wire right at t = 0, and a fully charged capacitor acts like an open circuit at steady state.

• After one time constant, a charging capacitor reaches about 63% of its maximum charge, and after two it reaches about 86.5%.

• The governing equation RC(dq/dt) + q = Cε comes straight from Kirchhoff's loop rule with I = dq/dt and V_C = q/C.

• During charging, half the energy from the battery is stored in the capacitor and half is dissipated in the resistor.

Frequently asked questions about RC Circuit

What is an RC circuit in AP Physics C E&M?

It's a circuit with a resistor and capacitor where the capacitor charges or discharges exponentially over time. Its speed is set by the time constant τ = RC, and you model it with the differential equation RC(dq/dt) + q = Cε.

Is the capacitor fully charged after one time constant?

No. After one τ it's only at about 63% of its maximum charge, and after two τ it's about 86.5%. Mathematically it never quite reaches 100%, but after about 5τ it's close enough to treat as fully charged.

Does a bigger resistor change how much charge the capacitor ends up with?

No. The final charge is Cε regardless of R. A bigger resistor only makes the charging take longer, because it raises τ = RC.

How is an RC circuit different from an LR circuit?

Both use exponential math, but the extremes flip. A capacitor starts like a wire and ends like an open circuit, while an inductor starts like an open circuit and ends like a wire. RC circuits use τ = RC, and LR circuits (Unit 13) use τ = L/R.

Are RC circuits on the AP Physics C E&M exam?

Yes. They're Topic 11.8 in Unit 11. Expect questions on initial current (ε/R), time-constant calculations, sketching q or I versus time, and deriving or rewriting the RC differential equation.