Problem 37 - College Physics

Question

In a series \(R-L-C\) circuit, the phase angle is \(40.0^{\circ},\) with the source voltage leading the current. The reactance of the capacitor is \(400 \Omega\), and the resistance of the resistor is \(200 \Omega .\) The average power delivered by the source is \(150 \mathrm{~W}\). Find (a) the reactance of the inductor, (b) the rms current, (c) the rms voltage of the source.

Step-by-Step Solution

Answer

(a) The reactance of the inductor is approximately 567 Ω. (b) The rms current is 0.866 A. (c) The rms voltage is about 277 V.

Step 1: Understanding the Phase Angle

The phase angle \( \theta \) is given as \( 40.0^{\circ} \). Since the source voltage leads the current, the circuit is predominantly inductive.

Step 2: Calculate the Total Impedance

We use the formula for impedance, \( Z = \sqrt{R^2 + (X_L - X_C)^2} \), where \( X_L \) is the inductive reactance and \( X_C \) is the capacitive reactance (given as \( 400 \Omega \)). The resistance \( R \) is \( 200 \Omega \).

Step 3: Calculate the Reactance of the Inductor

Using the phase relationship \( \tan(\theta) = \frac{X_L - X_C}{R} \), and knowing \( \tan(40^{\circ}) \approx 0.8391 \), we have:\[ X_L - 400 \Omega = 0.8391 \times 200 \Omega\]Solving gives \( X_L \approx 567 \Omega \).

Step 4: Calculate the RMS Current

Use the power formula \( P = I_{rms}^2 R \) to find \( I_{rms} \):\[ I_{rms} = \sqrt{\frac{P}{R}} = \sqrt{\frac{150 \, \text{W}}{200 \Omega}} \approx 0.866 \text{ A}\]

Step 5: Calculate the RMS Voltage

The voltage across the circuit is \( V_{rms} = I_{rms} \cdot Z \). First find \( Z \) using the inductive reactance:\[ Z = \sqrt{200^2 + (567 - 400)^2} \approx 320 \Omega\]Thus, \( V_{rms} \approx 0.866 \, \text{A} \times 320 \, \Omega \approx 277 \text{ V} \).

Key Concepts

Phase Angle

The phase angle in an RLC circuit is crucial for understanding how the circuit behaves. It tells us how much the current waveform is shifted compared to the voltage waveform. In this exercise, we are given a phase angle of \(40.0^{\circ}\) with the source voltage leading the current. This indicates that the circuit is more inductive in nature.

In inductive circuits, the current lags behind the voltage. This leads to a positive phase angle. The phase angle can be calculated using the tangent function:

• \( \tan(\theta) = \frac{X_L - X_C}{R} \)

where \(X_L\) is the inductive reactance and \(X_C\) is the capacitive reactance. You'll use this relationship whenever you need to find how the phase angle correlates with the other properties of the circuit like resistance or reactance.

Inductive Reactance

Inductive reactance, \(X_L\), is a measure of the opposition that an inductor provides to alternating current (AC). It depends on the frequency of the AC and the inductance of the coil, and it can be calculated using the formula:

• \( X_L = 2\pi fL \)

where \( f \) is the frequency and \( L \) is the inductance. In the context of this exercise, we're solving for the inductive reactance using the phase angle and other given values.

From the step-by-step solution, we use the equation for the phase angle \( \tan(\theta) = \frac{X_L - X_C}{R} \) to calculate \(X_L = 567 \Omega\). This shows how the inductor's reactance is contributing to the overall behavior of the circuit, especially its phase relationship.

RMS Current

RMS (Root Mean Square) current is a way of expressing AC current that conveys the equivalent DC current value. It provides a measure of the current's power-delivering capability. RMS values are preferred in AC circuits to calculate power.

In this exercise, the RMS current is determined using the average power delivered by the source. We use the relationship:

• \( P = I_{rms}^2 R \)

From the solution, the RMS current is computed as: \[ I_{rms} = \sqrt{\frac{150 \, \text{W}}{200 \, \Omega}} \approx 0.866 \, \text{A} \]This indicates how much current is effectively flowing through the resistor in the circuit, providing insights into the circuit's performance.

Impedance

Impedance, denoted by \( Z \), is a comprehensive measure that encompasses both resistance and reactance in an AC circuit. It provides an overall picture of how much the circuit resists current flow. The impedance formula is given by:

• \( Z = \sqrt{R^2 + (X_L - X_C)^2} \)

In this problem, the impedance combines the resistance \( R \) and the reactance difference \( X_L - X_C \). Calculations using these values show:\[ Z = \sqrt{200^2 + (567 - 400)^2} \approx 320 \, \Omega \]Understanding impedance is vital for predicting how voltage and current will behave in complex AC networks. It is also crucial when calculating the RMS voltage of the source by using \( V_{rms} = I_{rms} \times Z \), leading to approximately \( 277 \, \text{V} \) in this exercise.