An LC circuit consists of a capacitor, C = 4.08 µF, and an inductor, L = 4.92 mH

Question

An LC circuit consists of a capacitor, C = 4.08 µF, and an inductor, L = 4.92 mH. The capacitor is fully charged using a battery and then connected to the inductor. An oscilloscope is used to measure the frequency of the oscillations in the circuit. Next, the circuit is opened, and a resistor, R, is inserted in series with the inductor and the capacitor. The capacitor is again fully charged using the same battery and then connected to the circuit. The angular frequency of the damped oscillations in the RLC circuit is found to be 20.5% less than the angular frequency of the oscillations in the LC circuit.

a) Determine the resistance of the resistor.
b) How long after the capacitor is reconnected in the circuit will the amplitude of the damped current through the circuit be 34.5% of the initial amplitude?
c) How many complete damped oscillations will have occurred in that time?

Explanation / Answer

a)angular frequency without resistor = = 1/LC = 7075.56 Hz

angular frequency with resistor = (1/LC - R2/4L2) = 79.5% of 7075.56

solving gives r = 41.95

b)initial current amplitude = cv0

damped current amplitude = v0/Le-at where =  (1/LC - R2/4L2) and a= R/2L

now this is 34.5% of the initial amplitude.

so  v0/Le-at/  cv0 = 0.345

solving we can get t=

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