The questions are on the RC circuits test. Take it under real conditions before reading the mark scheme.
Quick answer: Kirchhoff's junction rule conserves charge and his loop rule conserves energy, giving equations for every branch current. This mark scheme gives the model answers and the marks for the 8 questions of the AP Physics C test on RC circuits, from Kirchhoff’s rules to sketching the current, with the equivalent answers to accept.
Award marks as shown. Accept answers within rounding, and carry forward an earlier slip without penalising it twice. Take off a mark once if a final answer has no unit.
Level: AP Physics C (Mechanics and E&M, calculus-based) · Curriculum: AP PC:E&M Unit 11
Question 1 Kirchhoff’s rules (2 marks)
(a) The junction rule expresses conservation of electric charge: charge cannot accumulate at a junction in steady state. (b) The loop rule expresses conservation of energy: the net change in potential around a closed loop is zero.
Marking: 1 mark for each correct principle.
Question 2 Battery and load (2 marks)
I = ε/(R + r) = 15/5.0 = 3.0 A. V = ε − Ir = 15 − 3.0 = 12 V.
Marking: 1 mark for the current; 1 mark for the terminal voltage.
Question 3 Charging a capacitor (3 marks)
(a) τ = RC = (5.0 × 10⁴)(20 × 10⁻⁶) = 1.0 s.
(b) VC = 10(1 − e⁻²) = 8.6 V.
(c) I = (10/5.0 × 10⁴)e⁻² = (2.0 × 10⁻⁴)(0.1353) = 27 μA.
Marking: 1 mark each part.
Question 4 Discharge time (3 marks)
τ = (1.0 × 10³)(470 × 10⁻⁶) = 0.47 s. 4.0 = 12e−t/τ gives t = τ ln 3 = (0.47)(1.0986) = 0.52 s.
Marking: 1 mark for τ; 1 mark for the equation; 1 mark for the answer with unit.
Question 5 Limits in time (2 marks)
At t = 0 it has no voltage across it, so it acts like a wire and the current is ε/R. After a long time its voltage equals ε, so the current is zero and it behaves like an open circuit.
Marking: 1 mark for each limit.
Question 6 Derive the equation (3 marks)
Loop rule: ε − R(dq/dt) − q/C = 0. With q = Cε(1 − e−t/RC), dq/dt = (ε/R)e−t/RC, so R(dq/dt) + q/C = εe−t/RC + ε(1 − e−t/RC) = ε, which satisfies the equation.
Marking: 1 mark for the equation; 1 mark for the derivative; 1 mark for the substitution.
Question 7 Energy balance (3 marks)
Battery: Cε² = (100 × 10⁻⁶)(36) = 3.6 mJ. Stored: ½Cε² = 1.8 mJ. Dissipated: 3.6 − 1.8 = 1.8 mJ.
Marking: 1 mark for each energy.
Question 8 Sketching the current (2 marks)
The sketch is a decaying exponential from I₀ = V₀/R at t = 0, concave up, approaching zero. At t = τ the current is marked at 0.37 I₀, where I₀e⁻¹ = 0.37 I₀.
Marking: 1 mark for the shape and intercept; 1 mark for τ marked at 37 %.
