Problem 75 - Fundamentals of Physics

Question

For a certain driven series \(R L C\) circuit, the maximum generator emf is \(125 \mathrm{~V}\) and the maximum current is \(3.20 \mathrm{~A}\). If the current leads the generator emf by 0.982 rad, what are the (a) impedance and (b) resistance of the circuit? (c) Is the circuit predominantly capacitive or inductive?

Step-by-Step Solution

Answer

(a) Impedance is 39.06 Ω, (b) Resistance is 21.81 Ω, (c) The circuit is predominantly capacitive.

Step 1: Identify Known Values

The given values are the maximum generator emf, \( E_{max} = 125 \text{ V} \), the maximum current, \( I_{max} = 3.20 \text{ A} \), and the phase angle \( \phi = 0.982 \text{ rad} \) through which the current leads the emf.

Step 2: Calculate Impedance

Impedance \( Z \) in an \( RLC \) circuit is calculated by the formula \( Z = \frac{E_{max}}{I_{max}} \). By plugging in the given values, we have:\[ Z = \frac{125 \text{ V}}{3.20 \text{ A}} = 39.06 \Omega \].

Step 3: Calculate Resistance

Use the phase angle to find the resistance. The phase angle is given by \( \tan \phi = \frac{X_L - X_C}{R} \) where \( X_L \) and \( X_C \) are inductive and capacitive reactances respectively. However, we can use the cosine formula: \( \cos \phi = \frac{R}{Z} \). Rearrange to find \( R \):\[ R = Z \cos \phi = 39.06 \cos(0.982) = 21.81 \Omega \].

Step 4: Determine Circuit Type

Since the current leads the voltage by \( 0.982 \text{ rad} \), the circuit is predominantly capacitive. The phase angle indicates that the capacitive reactance \( X_C \) is greater than the inductive reactance \( X_L \).

Key Concepts

Impedance

In an RL circuit, impedance is an essential concept that refers to the opposition to the flow of alternating current. Impedance, often represented by the symbol \( Z \), combines all resistive and reactive components in the circuit. It includes three parts: resistance \( R \), inductive reactance \( X_L \), and capacitive reactance \( X_C \).

Impedance is measured in ohms (\( \Omega \)). Its value can be calculated using the formula:

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

However, for simplicity when only resistance and impedance are considered, particularly in textbook problems, the impedance can be derived using the peak values of emf and current:

• \[ Z = \frac{E_{max}}{I_{max}} \]

Understanding impedance is crucial because it characterizes how the circuit responds to different frequencies of the alternating current, which in turn impacts the circuit's overall performance.

Resistance

Resistance is the part of a circuit that solely opposes current flow, without influencing the signal phase. It's an inherent property of materials used in the circuit, such as resistors, and is measured in ohms (\( \Omega \)).

In an RL circuit, resistance doesn’t change with frequency, unlike reactance, and it can be calculated if we know the impedance and phase angle. Use the formula:

• \[ R = Z \cos(\phi) \]

This formula signifies that with the known impedance and phase angle, one can easily find resistance. The role of resistance in an RL circuit is crucial as it determinately influences how much current flows for a given voltage. Managing resistance effectively ensures efficient circuit operation.

Phase Angle

The phase angle in an RL circuit indicates the difference in phase between the voltage across the circuit and the current through it. Typically denoted by \( \phi \) (phi), the phase angle explains whether the circuit is more inductive or capacitive.

When the phase angle is positive, inductive reactance is stronger, causing the current to lag behind the voltage. Conversely, a negative phase angle indicates a predominance of capacitive reactance, leading to the current leading the voltage. To compute resistance when known, the cosine of the phase angle, \( \cos(\phi) \), is used, demonstrating the relationship:

• \[ Z \cos(\phi) = R \]

Understanding the phase angle is crucial for determining the type of reactance affecting the circuit and the timing relations of current and voltage.

Capacitive and Inductive Reactance

Reactions in an RL circuit occur due to the presence of inductors and capacitors, giving rise to two types of reactance: inductive reactance \( X_L \) and capacitive reactance \( X_C \).

Inductive reactance occurs because inductors oppose changes in current, leading currents to lag behind voltages. It can be calculated using the formula:

• \[ X_L = 2\pi fL \]

where \( f \) is the frequency and \( L \) the inductance.

Capacitive reactance arises as capacitors oppose voltage changes, causing currents that lead voltages. It follows the formula:

• \[ X_C = \frac{1}{2\pi fC} \]

where \( C \) is the capacitance.

The net reactance of the circuit is the difference: \( X_L - X_C \). This distinction significantly impacts whether the circuit exhibits more inductive or capacitive properties, influencing the overall circuit behavior and phase relationship.