Answer to Question #116550 in Electricity and Magnetism for Joel George

Question

a) Plot currents through the capacitor and the inductance of a series RLC circuit at resonance on
the same graph as functions of time. (b) What is the phase difference between the current and the
total voltage at frequency w>w0 , such that current amplitude is Imax/√2 ?

Answer 1

a) The currents will be equal. Moreover, the magnitude of the current will be maximum:

b) Assume that we have voltage source V, resistance R, capacitance C, and inductance L. First, find the resonant frequency:

$$\omega_0=\frac{1}{\sqrt{LC}}<\omega.$$

The maximum value of current is at resonance:

$$I_\text{max}=\frac{V}R{}.$$

We need a current of

$$I=\frac{I_\text{max}}{\sqrt2}=\frac{V}{R\sqrt2}.$$

Thus, the impedance of a series RLC circuit is

$$Z=\sqrt{R^2+(X_L-X_C)^2}=R\sqrt2,$$

and since the phase difference is

$$\phi=\text{atan}\frac{X_L-X_C}{R},$$

all we need is to find $(X_L-X_C)$ from the equation above for impedance Z:

$$X_L-X_C=R, \\ \phi=\text{atan}\frac{R}{R}=0.785\text{ rad},$$

or 45°.