An RLCcircuit is driven by a generator with an emf amplitude ofand a current amplitude of $1.25A$. The current leads the emf by $\text{0.650rad}$. What are the (a) impedance and (b) resistance of the circuit? (c) Is the circuit inductive, capacitive, or in resonance.

emf amplitude voltage is 80 V.

The current amplitude is $1.25A$.

The phase angle is $0.650rad$.

The resistance in an AC circuit is called impedance. The resistance due to the capacitor in an AC circuit is called capacitive reactance and the resistance due to the inductor is called inductive reactance.

Use the formula for impedance in terms of amplitude emf voltage, and current, and then from that, find resistance.

The formulae are as follows:

$z=\frac{n}{i}······\left(1\right)$

Here, n is EMF amplitude voltage, i is current.

$Z=\sqrt{R^2+{\left(X_L-X_C\right)}^2}······\left(2\right)$

Here, Ris resistance, $X_L$is indictive reactance and$X_C$ is capacitive reactance.

Use equation (1) to calculate the impedance.

$$\begin{gathered} z=\frac{n}{i} \\ Z=\frac{\text{80}}{\text{1.25}} \\ Z=64\Omega \end{gathered}$$

Hence,impedance is $64\Omega$.

Use the equation (2) to calculate the resistance.

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

$\text{tan}\varphi =\frac{X_L-X_c}{R}$

$\text{tan}\left(\text{0.65}\right)=\frac{X_L-X_c}{R}$

$$\begin{gathered} X_L-X_C=R\text{tan}\left(\text{0.65}\right) \\ {X}_L-X_C=\text{0.76}R \end{gathered}$$

So,

$$\begin{gathered} Z=\sqrt{R^2+{\left(0.76R\right)}^2} \\ Z=1.26R \\ 64=1.26R \\ R=50.9\Omega \end{gathered}$$

Hence, resistance is $50.9\Omega$.

As current leads the voltage, the circuit is capacitive.

Hence, the circuit is capacitive.

Thus, find the impedance using the formula in terms of emf and current. Also, find the resistance using the formula for impedance in terms of resistance, and inductive and capacitive reactance. Using the lag between current and voltage, find that the circuit is capacitive.