The method and the results are on the RC circuits lab. Do the analysis yourself first, then check the answers below.
Quick answer: Kirchhoff's junction rule conserves charge and his loop rule conserves energy, giving equations for every branch current. These answers go with the AP Physics C lab on RC circuits: the graph, the calculations, the uncertainties and the conclusion explained, for all 7 analysis questions.
The answers use the sample values. Your own τ will depend on your actual capacitor, which can differ from its label by a large margin, but the log graph should still be a straight line and τ/R should land within the tolerance of the component.
Level: AP Physics C (Mechanics and E&M, calculus-based) · Curriculum: AP PC:E&M Unit 11
Question 1 Plot ln V against t
V = V₀e−t/τ, so ln V = ln V₀ − t/τ. The points lie on a straight line of negative gradient.
Question 2 Gradient
Gradient = (−1.35 − 1.79)/(60 − 0) = −3.14/60 = −0.052 s⁻¹.
Question 3 Time constant
τ = −1/gradient = 1/0.0523 = 19 s.
Question 4 Capacitance
C = τ/R = 19.1/22 000 = 8.7 × 10⁻⁴ F, so 870 μF, which is 13 % below the nominal value and within the usual 20 % tolerance of electrolytic capacitors.
Question 5 Half-time check
t½ = τ ln 2 = 19.1 × 0.693 = 13 s. The table shows V = 3.5 V at 10 s and 2.1 V at 20 s, so 3.0 V occurs between them, near 13 s, in agreement.
Question 6 Uncertainty
A 0.5 s timing error on 60 s is 0.8 %, but the small readings at 50–60 s have voltage uncertainty 0.01 V on 0.26 V, about 4 %, which affects the log slope most. Overall τ = 19 ± 1 s.
Question 7 Reducing error
Using a larger starting voltage and stopping at about 2τ avoids the noisy low readings, and recording with a video of the meter reduces reaction-time error. Both give a more reliable gradient.
