Quick answer: Kirchhoff's junction rule conserves charge and his loop rule conserves energy, giving equations for every branch current. This AP Physics C worksheet on RC circuits gives 20 problems from easy to hard: junction rule, time constant and start and finish. The worked answers are on a separate page, and the worksheet is a free printable PDF.
These twenty problems develop two linked skills: using Kirchhoff’s junction and loop rules to analyse multi-branch circuits, and using exponentials to follow a capacitor as it charges and discharges. They move from simple recall and one-step calculations to derivations, graph analysis and energy arguments in the style of AP Physics C: Electricity and Magnetism.
The set suits AP students, first-year university physics and A Level learners ready for calculus.
Level: AP Physics C (Mechanics and E&M, calculus-based) · Curriculum: AP PC:E&M Unit 11
Problem 1 Junction rule ★★★
At a junction 3.0 A and 2.0 A flow in, and 4.0 A flows out along one wire.1. State the junction rule.
2. Calculate the current in the remaining wire and say whether it enters or leaves.
Problem 2 Time constant ★★★
1. Calculate τ for R = 10 kΩ and C = 22 μF.2. Show that the units of RC are seconds.
Problem 3 Start and finish ★★★
Aarav charges a 100 μF capacitor through 1.5 kΩ from a 6.0 V battery.1. Calculate the current at t = 0.
2. State the current as t → ∞.
3. Calculate the final charge.
Problem 4 Fractions charged ★★★
1. State the fraction of the final charge present after t = τ.2. Calculate the fraction after 3τ.
3. Calculate the fraction after 5τ.
Problem 5 Discharge fraction ★★★
A capacitor with Q₀ = 80 μC discharges through a resistor.1. Calculate the charge remaining after 2τ.
2. State what fraction has gone.
Problem 6 Reading a charging graph ★★★
Chiamaka records VC against time while charging through a resistor.2. Calculate R if C = 1000 μF.
Problem 7 Resistors in a loop ★★★
A 12 V battery drives current through 4.0 Ω and 2.0 Ω in series.1. Use the loop rule to calculate the current.
2. Calculate the voltage across the 4.0 Ω resistor.
Problem 8 Two-loop circuit ★★★
Apply Kirchhoff’s rules to the circuit shown.2. Write the two loop equations.
3. Calculate I₁, I₂ and I₃, and say which battery is charging.
4. Check by comparing power supplied with power dissipated.
Problem 9 Steady state ★★★
A 12 V battery is in series with a 4.0 kΩ resistor, and then a 2.0 kΩ resistor in parallel with a 5.0 μF capacitor.1. Explain why no current flows in the capacitor branch after a long time.
2. Calculate the steady current.
3. Calculate the final voltage and charge on the capacitor.
Problem 10 Time to reach 8 V ★★★
Emma charges a 3.3 μF capacitor through 2.0 MΩ from 12 V.1. Calculate τ.
2. Calculate the time for VC to reach 8.0 V.
Problem 11 Discharge current ★★★
A 2200 μF capacitor at 9.0 V discharges through 470 Ω.1. Calculate τ.
2. Calculate the initial current.
3. Calculate the current after 2.0 s.
Problem 12 Half-time ★★★
A 10 μF capacitor discharges through 100 kΩ.1. Show that the half-time is τ ln 2.
2. Calculate it for this circuit.
3. Calculate the time for the charge to fall to one sixteenth.
Problem 13 Verify the charging law ★★★
Take q = Cε(1 − e−t/RC).1. Differentiate to find I.
2. Show that the loop equation ε = IR + q/C is satisfied.
Problem 14 Log graph of discharge ★★★
Bilal plots ln V against t for a discharging capacitor. The initial voltage is 12 V.2. Hence find τ.
3. Calculate R if C = 470 μF.
Problem 15 Battery with internal resistance ★★★
A 12 V battery of internal resistance 0.50 Ω drives a 5.5 Ω load.1. Calculate the current.
2. Calculate the terminal voltage.
Problem 16 Energy in charging ★★★
A 100 μF capacitor is charged from 12 V through a resistor R.1. Calculate the energy stored.
2. Integrate I²R to show the resistor dissipates the same energy.
3. Explain why the answer does not depend on R.
Problem 17 Divider charging ★★★
A 12 V source feeds a 10 kΩ resistor to a node, where a 10 μF capacitor is in parallel with a 20 kΩ resistor to ground.1. Calculate the final capacitor voltage.
2. Calculate the time constant using the equivalent resistance.
3. Calculate the time for the voltage to reach 6.0 V.
Problem 18 Leaky capacitor ★★★
A parallel-plate capacitor is filled with a material of dielectric constant 4.0 and resistivity 1.0 × 10¹² Ω·m.1. Show that the leakage time constant is ρκε₀ whatever the plate size.
2. Calculate it (ε₀ = 8.85 × 10⁻¹² F/m).
Problem 19 Flash lamp ★★★
Kofi’s flash circuit has a 470 μF capacitor charged to 25 V that discharges through a 100 Ω lamp.1. Calculate the initial energy.
2. Show that the energy dissipated up to time t is U₀(1 − e−2t/τ).
3. Calculate the energy dissipated in the first τ.
Problem 20 Spot the error ★★★
A student says: ‘The capacitor reaches 63 % of full charge in τ, so it will be full after 1.6τ.’ The circuit has ε = 6.0 V, R = 10 kΩ and τ = 0.22 s.1. Explain the flaw.
2. Calculate the time to reach 99 %.
3. Calculate the current at that moment.
