Question

An ac series circuit containing a capacitor, inductor, and resistance is found to have a current of amplitude \(0.50 \mathrm{A}\) for a source voltage of amplitude \(10.0 \mathrm{V}\) at an angular frequency of $200.0 \mathrm{rad} / \mathrm{s} .\( The total resistance in the circuit is \)15.0 \Omega$ (a) What are the power factor and the phase angle for the circuit? (b) Can you determine whether the current leads or lags the source voltage? Explain.

Short answer

Based on the given information and calculations for this circuit: a) The power factor is 0.75, and the phase angle between the voltage and current is 41.4 degrees. b) It is not possible to determine whether the current leads or lags the source voltage without more information about the inductor and capacitor values in the circuit.

Step-by-step solution

Calculate impedance amplitude

The Impedance (Z) of the circuit can be found using the given amplitude of Voltage(V) and Current(I) using, \(Z =\frac{V}{I}\). So, plug in the given values to get the Impedance (Z) of the circuit. $$Z = \frac{10.0 V}{0.50 A} = 20.0 \,\Omega !$$

Calculate reactance amplitude

Next, we find the amplitude of the reactance (X). We are given the resistance (R) in the circuit and we have calculated impedance (Z), so we can use the following formula to find the reactance (X): $$X = \sqrt{Z^2 - R^2}$$ where X is the reactance, Z is impedance, and R is resistance. $$X = \sqrt{(20.0 \, \Omega)^2 - (15.0 \, \Omega)^2} = \sqrt{625} \, \Omega$$ $$X = 25.0 \, \Omega$$

Calculate the power factor

Now that we have the impedance (Z) and resistance (R), we can calculate the power factor (PF) of the circuit using the following formula: $$PF = \frac{R}{Z}$$ $$PF = \frac{15.0 \, \Omega}{20.0 \, \Omega} = 0.75$$

Calculate the phase angle

Using the power factor, we can calculate the phase angle (θ) between the voltage and current using: $$\cos{\theta} = PF$$ $$\theta = \arccos{(PF)}$$ $$\theta = \arccos{(0.75)} = 41.4°$$

Determine if current leads or lags the source voltage

The circuit contains a resistor, inductor, and capacitor. In an inductor, the current lags the voltage, and in a capacitor, the current leads the voltage. Since we are not told the values of the inductor (L) and capacitor (C), we cannot determine the net effect in this case. However, if the net effect was reactive (inductive or capacitive), the magnitude of the phase angle would have been exactly 90°. But the phase angle is 41.4° which is not exactly 90°. So, there must be some complex combination of inductor and capacitor effects that lead to a net phase angle of 41.4°. Hence, we cannot determine whether the current leads or lags the source voltage without more information. In conclusion: a) The power factor for the circuit is 0.75, and the phase angle is 41.4°. b) We cannot determine whether the current leads or lags the source voltage without more information on the inductor and capacitor values.