Asinusoidal voltage δv = 35.0 sin 100t, where δv is in volts and t is in seconds, is applied to a series rlc circuit with l = 165 mh, c = 99.0 µf, and r = 66.0 ω. (a) what is the impedance of the circuit? ω (b) what is the maximum current? a (c) determine the numerical value for ω in the equation i = imax sin (ωt − ϕ). rad/s (d) determine the numerical value for ϕ in the equation i = imax sin (ωt − ϕ).

(a) 107.2 Ω

The voltage in the circuit is written in the form

where in this case

V0 = 35.0 V is the maximum voltage

is the angular frequency

Given the inductance: L = 165 mH = 16.5 H, the reactance of the inductance is:

Given the capacitance: , the reactance of the capacitor is:

And given the resistance in the circuit, R = 66.0 Ω, we can now find the impedance of the circuit:

(b) 0.33 A

The maximum current can be calculated by using Ohm's Law for RLC circuit. In fact, we know the maximum voltage:

V0 = 35.0 V

The equivalent of Ohm's law for an RLC circuit is

where

Z=107.2 Ω

is the impedance. Substituting these values into the formula, we find:

(c) 100 rad/s

The equation for the current is

where

I0 = 0.33 A is the maximum current, which we have calculated previously

is the angular frequency

is the phase shift

In an RLC circuit, the voltage and the current have the same frequency. Therefore, we can say that

also for the current.

(d) -0.907 rad

Here we want to calculate the phase shift , which represents the phase shift of the current with respect to the voltage. This can be calculated by using the equation:

Substituting the values that we found in part a), we get

And the sign is negative, since the capacitive reactance is larger than the inductive reactance in this case. Converting in radians,

So the complete equation of the current would be

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