The questions are on the capacitance test. Take it under real conditions before reading the mark scheme.
Quick answer: Capacitance is the charge stored per unit potential difference, C = Q/ΔV, measured in farads. This mark scheme gives the model answers and the marks for the 8 questions of the AP Physics C test on capacitance, from capacitance and charge to series pair, with the equivalent answers to accept.
Award marks as shown. Accept results within rounding, give full credit to a correct method carried through from an earlier slip, and take off a mark once for a missing unit.
Level: AP Physics C (Mechanics and E&M, calculus-based) · Curriculum: AP PC:E&M Unit 10
Question 1 Capacitance and charge (2 marks)
Capacitance is the ratio C = Q/ΔV of the charge on one conductor to the potential difference between them. Since ΔV is proportional to Q for a fixed geometry, the ratio is constant and depends only on shape and material.
Marking: 1 mark for the definition; 1 mark for the proportionality argument.
Question 2 Parallel plate capacitor (3 marks)
C = ε₀A/d = (8.85 × 10⁻¹²)(0.050)/(1.0 × 10⁻³) = 4.4 × 10⁻¹⁰ F. Q = CΔV = (4.43 × 10⁻¹⁰)(9.0) = 4.0 × 10⁻⁹ C.
Marking: 1 mark for the formula and substitution; 1 mark for C; 1 mark for Q.
Question 3 Energy stored (3 marks)
U = ½CΔV² = ½(220 × 10⁻⁶)(900) = 0.099 J. P = U/t = 0.099/(5.0 × 10⁻⁴) = 2.0 × 10² W.
Marking: 1 mark for the formula; 1 mark for U; 1 mark for P.
Question 4 Dielectric inserted (3 marks)
Q = 5.0 nC is fixed. V = 50/2.5 = 20 V, and U = ½QV = ½(5.0 × 10⁻⁹)(20) = 5.0 × 10⁻⁸ J.
Marking: 1 mark for Q constant; 1 mark for V; 1 mark for U.
Question 5 Why the field falls (2 marks)
The dielectric polarises: its surface facing the positive plate gains negative bound charge and the other face gains positive bound charge. These oppose the field of the plates, so the net field and potential difference fall.
Marking: 1 mark for polarisation producing bound charge; 1 mark for the opposing field reducing E.
Question 6 Coaxial capacitor (3 marks)
C = 2πε₀L/ln(b/a) = (5.56 × 10⁻¹¹)(2.0)/ln 4 = 1.112 × 10⁻¹⁰/1.386 = 8.0 × 10⁻¹¹ F.
Marking: 1 mark for the formula; 1 mark for ln 4; 1 mark for the value with unit.
Question 7 Energy density (2 marks)
u = ½ε₀E² = ½(8.85 × 10⁻¹²)(1.2 × 10⁴)² = 6.4 × 10⁻⁴ J/m³.
Marking: 1 mark for the formula; 1 mark for the answer with unit.
Question 8 Series pair (2 marks)
a. 1/C = 1/6.0 + 1/3.0 = 1/2.0, so C = 2.0 μF.
b. Q = (2.0 μF)(9.0 V) = 18 μC, so V = 18/3.0 = 6.0 V.
Marking: 1 mark for each part.
