A coil of resistance and inductance is connected in series with a capacitor and an AC…

PHYSICS · Subjective · A Level · Physics · Fields and their consequences · Fields and their consequences

Problem

A coil of resistance

R

and inductance

L

is connected in series with a capacitor

C

and an AC source of fixed amplitude. The circuit shows a pronounced resonance at angular frequency

\omega_0

. (a) Starting from the expression for the current amplitude in the circuit, obtain the condition on impedance when the current amplitude reduces to

\dfrac{1}{\sqrt{2}}

of its maximum value. (b) Hence derive the quadratic equation satisfied by the two angular frequencies

\omega_1

and

\omega_2

at which this reduction occurs. (c) Solve the quadratic to obtain expressions for

\omega_1

and

\omega_2

in terms of

R,\,L,

and

C

. (d) Using the results of part (c), show that when

R

is much smaller than

\sqrt{\dfrac{L}{C}}

, the bandwidth

\Delta\omega = \omega_2-\omega_1

can be approximated by

\dfrac{R}{L}