A student investigates the discharge of a capacitor in the circuit shown in Fig. 2.1. E C V R1 R2 P Q Fig. 2.1 The student closes the switch and charges the capacitor. The switch is opened and a stop-watch is started. The capacitor discharges through the two resistors of resistance R₁ and R₂ connected between P and Q. At a fixed time t the potential difference V across the capacitor is measured. The experiment is repeated for different values of R₁ and R₂. It is suggested that V, R₁ and R₂ are related by the equation V t In( ) = - E C(R₁ + R₂) where E is the electromotive force (e.m.f.) of the battery and C is the capacitance of the capacitor.

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2021). The mark or mark type is shown in brackets.

1. [1] gradient = -t/C
2. [1] y-intercept = In E
3. [1] Values of (R₁ + R₂)/kΩ and 1/(R₁ + R₂)/10⁻⁶Ω⁻¹ correct as shown below: (R₁+R₂)/kΩ: 55 (±3), 1/(R₁+R₂)/10⁻⁶Ω⁻¹: 18 or 18.2 ± 0.9 (R₁+R₂)/kΩ: 69 (± 3 or 4), 1/(R₁+R₂)/10⁻⁶Ω⁻¹: 14 or 14.5 ± 0.7 (R₁+R₂)/kΩ: 90 (± 4 or 5), 1/(R₁+R₂)/10⁻⁶Ω⁻¹: 11 or 11.1 ± 0.6 (R₁+R₂)/kΩ: 80 (±4), 1/(R₁+R₂)/10⁻⁶Ω⁻¹: 13 or 12.5 ± 0.6 (R₁+R₂)/kΩ: 101 (±5), 1/(R₁+R₂)/10⁻⁶Ω⁻¹: 9.9 or 9.90 or 9.901 ± 0.5 (R₁+R₂)/kΩ: 115 (±6), 1/(R₁+R₂)/10⁻⁶Ω⁻¹: 8.7 or 8.70 or 8.696 ± 0.4
4. [1] Absolute uncertainties in 1/(R₁ + R₂) from ± 0.9 or ± 1 to ± 0.4 or ± 0.5.
5. [1] Six points plotted correctly. Must be accurate to half a small square. Diameter of points must be less than half a small square.
6. [1] Error bars in 1/(R₁ + R₂) plotted correctly. All error bars must be plotted. Total length of bar must be accurate to less than half a small square and symmetrical.
7. [1] Line of best fit drawn. Points must be balanced. Do not accept line from top point to bottom point. Line must pass between (10.2, 1.10) and (10.8, 1.10) and between (16.7, 0.40) and (17.2, 0.40).
8. [1] Worst acceptable line drawn (steepest or shallowest possible line that passes through all error bars). All error bars must be plotted.
9. [1] Negative gradient determined with clear substitution of data points into Δy/Δx. Distance between data points must be at least half the length of the drawn line.
10. [1] Gradient of worst acceptable line determined. uncertainty = (gradient of line of best fit – gradient of worst acceptable line) or uncertainty = ½ (steepest worst line gradient – shallowest worst line gradient)
11. [1] y-intercept determined by substitution of point on line into y = mx + c.
12. [1] y-intercept of worst acceptable line determined by substitution of point on line into y = mx + c. uncertainty = y-intercept of line of best fit – y-intercept of worst acceptable line or uncertainty = ½ (steepest worst line y-intercept – shallowest worst line y-intercept) Do not accept ECF from false origin method.
13. [1] C determined using gradient and C and E both given to two or three significant figures. C = -t/gradient = 60/(c)(iii)
14. [1] E determined using y-intercept and C and E both given with correct SI unit. E = e^(y-intercept) unit of C: F or CV⁻¹ or sΩ⁻¹ unit of E: V
15. [1] Percentage uncertainty determined with method shown. percentage uncertainty = (|Δt/t| + |Δgradient/gradient|) × 100 Clear substitution must be shown for maximum/minimum methods.
16. [1] (R₁ + R₂) determined to at least two significant figures from (d)(i) or (c)(iii) and (c)(iv) with correct substitution including signs and correct power of ten(s). Do not accept ECF for POT from (c)(iii), (c)(iv) or (d). (R₁+R₂) = -t/C × 1/(In V - In E) = -60/C × 1/(In 5.0 - In E) or (R₁+R₂) = gradient / (In 5.0 - y-intercept) = (c)(iii) / (1.61 - (c)(iv))