Physics

Problem

A student assembles a series RLC circuit with a resistor of resistance

R = 20.0 \pm 0.5\;\Omega

and a capacitor of capacitance

C = 0.050\;\mu\text{F}

(negligible error). Using a variable–frequency a.c. source of fixed r.m.s. voltage 8.0 V, the maximum current is observed at a frequency of

f_0 = 3.20 \pm 0.05\;\text{kHz}

. a. (i) Calculate the inductance of the coil and the percentage uncertainty in its value arising from the uncertainty in the measured resonant frequency. b. (ii) Determine the r.m.s. current at resonance and the percentage uncertainty in this current. c. (iii) When the frequency is increased to 3.50 kHz the r.m.s. current falls to 0.30 A. Using your component values, show quantitatively whether the circuit is predominantly inductive or capacitive at 3.50 kHz.

Intuitive explanation

At resonance in an RLC circuit, the inductive and capacitive reactances cancel out, leaving only resistance to oppose the current—this is why maximum current flows at a specific frequency. You can use the resonance condition $f_0 = \frac{1}{2\pi\sqrt{LC}}$ to find the unknown inductance $L$ , then propagate the uncertainties in $R$ and $f_0$ through your calculations to determine the final error in whatever quantity you're asked to find.

Intuitive explanation

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