A 220 μF capacitor charges through an unknown resistor from a 12.0 V ideal supply. Recorded (time in s, capacitor voltage in V) pairs are (0, 0), (1, 4.72), (2, 7.59), (3, 9.32), and (4, 10.38). Determine the time constant and resistance, the charge at 4.0 s, and the current at 2.0 s. Explain a linearized graph that could test the exponential model and how voltmeter input resistance could bias the result.

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τ ≈ 2.00 s, R ≈ 9.09 kΩ, q(4 s) = 2.28 mC, and i(2 s) = 0.486 mA. Plot ln(12−V_C) against t; its gradient should be −1/τ. For charging, V_C = ε(1−e^(−t/τ)). At t = τ, V_C = 0.632ε = 7.58 V. The 2.0 s reading is 7.59 V, so τ ≈ 2.00 s. Since τ = RC, R = 2.00/(220 × 10⁻⁶) = 9.09 × 10³ Ω. The units seconds per farad reduce to ohms. At 4.0 s the charge is q = CV_C = (220 × 10⁻⁶)(10.38) = 2.28 × 10⁻³ C. Current is i = (ε/R)e^(−t/τ). At 2.0 s this is (12.0/9090)e^(−1) = 4.86 × 10⁻⁴ A. Rearrange to ln(ε−V_C) = ln ε − t/τ; a straight line tests the exponential model. A finite voltmeter resistance in parallel with the capacitor provides leakage, lowering the final voltage and distorting the inferred time constant unless included in the effective circuit.

Extract an RC time constant from charging data

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Capacitor charging observations

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Capacitor charging observations — supplied values
Time / s	Capacitor voltage / V
0	0
1	4.72
2	7.59
3	9.32
4	10.38