A coil of inductance 88 mH and unknown resistance and a 0.94 microF capacitor are connected in series with an alternating emf of frequency 930 Hz. If the phase angle between the applied voltage and current is 75 degrees, what is the resistance of the coil?
If the emf has magnitude 50 V, what is the current in the circuit?
Borrowed from HRW 33-56P.
Here the resistance has not been stated; the phase angle has been stated instead and must be used to find the value of the resistance.
First evaluate the reactances, remembering to convert the linear frequency 930 Hz to an angular frequency 5840 rad/sec. The inductive reactance is then 514 ohm and the capacitive reactance is 182 ohm for a total reactance in the circuit of 332 ohm from the difference. The circuit is thus inductive at this frequency, in agreement with the positive phase angle reported.
The tangent of the phase angle is 3.73 and should equal the ratio of total reactance to resistance. Solving for resistance, R = 89 ohm.
For the magnitude of the current, we need the impedance. Using the formula for the impedance of RLC series circuit, Z = 344 ohm. The current magnitude is (50 V) / (344 ohm) = 145 mA.