In an RLC circuit, these three clements are connected in series: a resistor of \(20.0 \Omega,\) a \(35.0-\mathrm{mH}\) inductor, and a 50.0 - \(\mu\) F capacitor. The ac source of the circuit has an rms voltage of \(100.0 \mathrm{V}\) and an angular frequency of $1.0 \times 10^{3} \mathrm{rad} / \mathrm{s} .$ Find (a) the reactances of the capacitor and inductor, (b) the impedance, (c) the rms current, (d) the current amplitude, (e) the phase angle, and (f) the rims voltages across each of the circuit elements. (g) Does the current lead or lag the voltage? (h) Draw a phasor diagram.

First, we need to determine the reactances of the capacitor (X_C) and the inductor (X_L). For a capacitor, the reactance is given by the formula: \(X_{C} = \dfrac{1}{\omega C}\) For an inductor, the reactance is given by the formula: \(X_{L} = \omega L\) Using the given angular frequency (\(\omega = 1000 \,\mathrm{ rad/s}\)), inductor value (\(L = 35.0 \,\mathrm{mH}\)), and capacitor value (\(C = 50.0 \,\mathrm{\mu F}\)), we can calculate the reactances.

Next, we need to determine the impedance (Z) of the RLC circuit, which is the combination of resistance and reactance. Impedance can be found using the formula: \(Z = \sqrt{R^2 + (X_{L} - X_{C})^2}\) With the calculated reactances and given resistor value (\(R = 20.0 \, \Omega\)), we can determine the impedance.

Now, we can find the rms current (I_rms) in the circuit, given by the formula: \(I_{\text{rms}} = \dfrac{V_{\text{rms}}}{Z}\) With the impedance and given rms voltage (\(V_{\text{rms}} = 100.0 \, \mathrm{V}\)), we can compute the rms current.

In an AC circuit, the amplitude of the current (I_0) is given by the following formula: \(I_{0} = \sqrt{2} \cdot I_{\text{rms}}\) Using the rms current computed in the previous step, we can find the amplitude of the current.

The phase angle (\(\phi\)) is the angle between the voltage and current in the circuit, determined from the resistive and reactive components. It can be calculated using the formula: \(\phi = \arctan{\left(\dfrac{X_{L} - X_{C}}{R}\right)}\) Using the calculated reactances and resistance, we can find the phase angle.

The rms voltages across the resistor (V_R), capacitor (V_C), and inductor (V_L) can be calculated using the following formulae: - \(V_R = I_{\text{rms}} \cdot R\) - \(V_C = I_{\text{rms}} \cdot X_C\) - \(V_L = I_{\text{rms}} \cdot X_L\) Using the rms current and the previously calculated resistance and reactances, we can find the rms voltages.

If the phase angle is positive, the current lags the voltage. If the phase angle is negative, the current leads the voltage. By examining the calculated phase angle, we can answer this part.

A phasor diagram visually represents the relationship between voltage and current in the AC circuit concerning the resistor, capacitor, and inductor. Here are the steps to draw the phasor diagram for the given circuit: 1. Draw a horizontal axis labeled as the real axis or resistance (\(R\)). 2. Draw a vertical axis labeled as the imaginary axis or reactance (\(X_L - X_C\)). 3. Plot the phasors (arrows) in the following order and direction: - \(V_R\) along the positive real axis (horizontal). - \(V_C\) along the negative imaginary axis (downwards). - \(V_L\) along the positive imaginary axis (upwards). 4. Add the phasors tip-to-tail to represent voltage and current (angle between them represents phase angle).