Problem 33 - College Physics

Question

A series \(R C\) circuit has a resistance of \(100 \Omega\) and a capacitive reactance of \(50 \Omega\). (a) Will the phase angle be (1) positive, (2) zero, or (3) negative? Why? (b) What is the phase angle of this circuit?

Step-by-Step Solution

Answer

(a) Positive, due to current leading voltage in R-C circuit. (b) Phase angle is approximately 26.57 degrees.

Step 1: Understanding the concept of phase angle

In an R-C circuit, the phase angle \( \phi \) is the angle between the voltage across the entire circuit and the current through it. A phase angle is positive if the current leads the voltage (typical in a capacitive circuit), zero if there's no phase shift, and negative if the current lags the voltage (typical in an inductive circuit).

Step 2: Determining the nature of the phase angle

In this circuit, the capacitive reactance \( X_C = 50 \Omega \) indicates that there is a capacitance component. In a pure R-C circuit, the current leads the voltage due to the capacitive nature, resulting in a positive phase angle.

Step 3: Calculating the phase angle

The phase angle \( \phi \) is calculated using the formula: \( \tan \phi = \frac{X_C}{R} \). Here, \( X_C = 50 \Omega \) and \( R = 100 \Omega \), thus:\[\tan \phi = \frac{50}{100} = 0.5\]To find \( \phi \), take the arctangent:\[\phi = \tan^{-1}(0.5)\]

Step 4: Computing and interpreting the phase angle

Calculate \( \phi = \tan^{-1}(0.5) \), which approximately equals 26.57 degrees. This phase angle is positive which indicates that in this R-C circuit, the current leads the voltage.

Key Concepts

Phase Angle

In an RC circuit, the phase angle is a crucial concept that describes the relationship between the voltage and the current. The phase angle, often symbolized as \( \phi \), is defined as the angle by which the current either leads or lags behind the voltage. In the context of RC circuits, the phase angle can tell us a lot about how the circuit behaves.

• A positive phase angle means that the current leads the voltage, which is typical in circuits with a capacitive component.
• If the phase angle is zero, it indicates that there is no phase difference, meaning the voltage and current are in perfect sync.
• A negative phase angle indicates that the current lags behind the voltage, usually found in circuits with inductive components.

In this particular exercise, we are dealing with an RC circuit where the phase angle is positive. This is due to the presence of capacitive reactance, which causes the current to lead the voltage. Thus, for an RC circuit like the one given, the phase angle is positive.

Capacitive Reactance

Capacitive reactance, denoted as \( X_C \), plays a significant role in an RC circuit. It is the opposition that a capacitor presents to the change in voltage in a circuit. The capacitive reactance can be thought of as a resistor, whose resistance depends inversely on the frequency of the current and the capacitance itself. The formula for capacitive reactance is: \[ X_C = \frac{1}{2\pi fC} \] where:

• \( f \) is the frequency of the input signal.
• \( C \) is the capacitance of the circuit.

For our particular case, we have a capacitive reactance of \( 50 \Omega \). This indicates quite a significant opposition to the current, although not zero. Because of the capacitive reactance, as defined earlier, this circuit will have a positive phase angle, since capacitive reactance causes the current to "lead" the voltage. Understanding capacitive reactance is key to analyzing how the current and voltage behave in the circuit.

Resistance

Resistance, often symbolized by \( R \), is one of the most fundamental concepts in any electrical circuit. In an RC circuit, resistance works together with capacitive reactance to determine the total impedance of the circuit. Impedance is the overall opposition to the flow of current, and it's a combination of both resistive and reactive elements.

• Resistance is measured in ohms (\( \Omega \)).
• In the given problem, the resistance is \( 100 \Omega \).

This resistance doesn't change with frequency, unlike reactance. It provides a constant opposition to current flow. In combination with capacitive reactance, it determines how much the current leads the voltage. The value of resistance directly influences the calculation of the phase angle, which is determined using the formula: \( \tan \phi = \frac{X_C}{R} \). Here, a higher resistance compared to the reactance results in a smaller phase angle. Thus, in analyzing RC circuits, knowing the resistance helps predict how the circuit will respond to alternating current, influencing both phase angle and overall circuit behavior.