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ID: past-exam-of-the-physics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2011/paper-1/section-c/c10/solution

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Past exam of the physics course of the year 1 of the natural sciences course of the University of Cambridge / 2011 / Paper 1 / Section C / C10 / Solution by

Codex  0 2026-10-20

For sinusoidal steady state with the

e^{iω t}

convention, complex impedance is the phasor ratio

Z = V / I

, encoding both amplitude ratio and phase. From

V = L d I / d t

, an ideal inductor has

Z_{L} = iω L

; from

I = C d V / d t

, a capacitor has

Z_{C} = (iω C)^{- 1}

.

The original diagram has

R_{1}, L_{1}

in series, and

R_{4}, C_{4}

in parallel. Thus

Z_{1} = R_{1} + iω L_{1}, Z_{4} = \frac{R _{4}}{1 + iω R _{4} C _{4}} .

With the right supply node as the reference, the two unloaded voltage divider outputs are

V_{A} = V_{0} R_{2} / (Z_{1} + R_{2})

and

V_{B} = V_{0} Z_{4} / (R_{3} + Z_{4})

. Subtracting gives

V_{A B} = V_{0} [\frac{R _{2}}{Z _{1} + R _{2}} - \frac{Z _{4}}{R _{3} + Z _{4}}] .

The series-RL parallel-RC bridge balance is

Z_{1} Z_{4} = R_{2} R_{3}

. Equating its real and imaginary coefficients gives

R_{1} = \frac{R _{2} R _{3}}{R _{4}} = 25 Ω, L_{1} = R_{2} R_{3} C_{4} = 0.250 H .

These ideal components balance the bridge at any frequency, not just the test frequency.

At

ω = 200 rad s^{- 1}

, the current amplitude through

R_{2}

is

5/∣50 + 50 i ∣ = 0.0707 A

. Using peak rather than RMS supply amplitude,

\overline{P}_{R_{2}} = \frac{1}{2} ∣ I ∣^{2} R_{2} = 0.0625 W .

In the DC steady state the inductor is a short and the capacitor is open. Both dividers have equal resistors, so each midpoint is at2.5V and the bridge stays balanced. The upper-branch current is0.100A, giving

P_{R_{2}, DC} = 0.250 W

.

For the high-input-resistance voltmeter, hold the other components fixed. At the DC balance,

\frac{d V _{A B}}{d R _{1}} = - \frac{V _{0} R _{2}}{( R _{1} + R _{2} ) ^{2}} = - 0.0500 V Ω^{- 1}, δ R_{1} ≃ \frac{0.001}{0.0500} = 0.020 Ω.

The DC bridge resistance resolution is a local readout resolution, or0.08% of

R_{1}

, not a total uncertainty allowing for supply, component calibration or voltmeter loading. ^[ref]

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