Problem 51 - Physics

Question

Part \(a\) of the drawing shows a resistor and a charged capacitor wired in series. When the switch is closed, the capacitor discharges as charge moves from one plate to the other. Part \(b\) shows a plot of the amount of charge remaining on each plate of the capacitor as a function of time. (a) What does the time constant \(\tau\) of this resistor-capacitor circuit physically represent? (b) How is the time constant related to the resistance \(R\) and the capacitance \(C ?(\mathrm{c})\) In part \(c\) of the drawing, the switch has been removed and an ac generator has been inserted into the circuit. What is the impedance \(Z\) of this circuit? Express your answer in terms of the resistance \(R\), the time constant \(\tau\), and the frequency \(f\) of the generator. Problem The circuit elements in the drawing have the following values: \(R=18 \Omega\) \(V_{\mathrm{rms}}=24 \mathrm{~V}\) for the generator, and \(f=380 \mathrm{~Hz}\). The time constant for the circuit is \(\tau=3.0 \times 10^{-4} \mathrm{~s}\). What is the rms current in the circuit?

Step-by-Step Solution

Answer

The time constant \( \tau \) is 0.3 ms, and the rms current is approximately 0.158 A.

Step 1: Understanding the Time Constant

The time constant \( \tau \) in a resistor-capacitor (RC) circuit is a measure of the time it takes for the charge on the capacitor to decrease to approximately \( \frac{1}{e} \) (about 36.8%) of its initial value during discharge. Physically, \( \tau \) represents how quickly the capacitor discharges through the resistor.

Step 2: Relationship of Time Constant to Resistance and Capacitance

The time constant \( \tau \) of an RC circuit is given by the formula \( \tau = R \cdot C \), where \( R \) is the resistance and \( C \) is the capacitance. This indicates that the time constant is directly proportional to both the resistance and the capacitance of the circuit.

Step 3: Calculating Impedance

With the AC generator inserted, the impedance \( Z \) of the RC circuit combines resistance and capacitive reactance. The capacitive reactance is \( X_c = \frac{1}{2\pi f C} \). The impedance is then given by \( Z = \sqrt{R^2 + X_c^2} \). Substitute \( C = \frac{\tau}{R} \) obtained from \( \tau = RC \) into the formula for \( X_c \): \( X_c = \frac{1}{2\pi f (\frac{\tau}{R})} = \frac{R}{2\pi f \tau} \). Plug this expression into the impedance formula.

Step 4: Expressing Impedance in Terms of Given Variables

From Step 3, we have \( Z = \sqrt{R^2 + (\frac{R}{2\pi f \tau})^2} \). Simplify this into a single expression to use in subsequent calculations.

Step 5: Solving for RMS Current

Use Ohm's Law for AC circuits, \( I_{\mathrm{rms}} = \frac{V_{\mathrm{rms}}}{Z} \). Substitute \( V_{\mathrm{rms}} = 24 \mathrm{~V} \), \( R = 18 \Omega \), \( f = 380 \mathrm{~Hz} \), and \( \tau = 3.0 \times 10^{-4} \mathrm{~s} \) into the expression for \( Z \), then calculate \( I_{\mathrm{rms}} \).

Step 6: Final Computation of RMS Current

Calculate \( X_c = \frac{18}{2\pi \times 380 \times 3.0\times10^{-4}} \) to get \( X_c \approx 150.29 \Omega \). Then, \( Z = \sqrt{18^2 + 150.29^2} \approx 151.54 \Omega \). Finally, \( I_{\mathrm{rms}} = \frac{24}{151.54} \approx 0.158 \mathrm{~A} \).

Key Concepts

Time Constant

In an RC circuit, the time constant, often denoted by the Greek letter \( \tau \), is a crucial parameter. It demonstrates how quickly the capacitor in the circuit charges or discharges. When a resistor and a capacitor are placed in series, the time constant indicates the period required for the capacitor to lose about 63.2% of its stored charge during discharge. This value comes from the mathematical constant \( e \), where \( \tau \) represents the time it takes for the charge to fall to \( \frac{1}{e} \) of its initial value. After one time constant, the remaining charge on the capacitor is 36.8% of the initial charge. This behavior characterizes the rate at which current in the circuit decreases exponentially over time, making the time constant a critical factor in understanding RC circuits.

Impedance

Impedance, symbolized as \( Z \), is a broad concept that includes both the resistance and reactance in an electrical circuit. In the context of an RC circuit with an AC generator, the impedance merges the pure resistance with the capacitive reactance, taking into account the circuit's resistance and its response to changing voltages or currents. The formula for impedance in an RC circuit with a resistor \( R \) and capacitive reactance \( X_c \) is \( Z = \sqrt{R^2 + X_c^2} \). The capacitive reactance \( X_c \) is dependent on the frequency and capacitance, given by the equation \( X_c = \frac{1}{2\pi f C} \). This impedance is an essential factor in AC circuits as it influences how the circuit responds to alternating current, affecting the amplitude and phase of voltage and current.

Capacitance

Capacitance is a measure of a capacitor's ability to store energy as electrical charge. It is denoted by \( C \) and is typically measured in farads (F). In an RC circuit, capacitance along with resistance determines the rate at which a capacitor charges and discharges. The relationship between capacitance, resistance, and the time constant \( \tau \) is \( \tau = R \cdot C \). This equation underscores that a higher capacitance will result in a longer time constant, meaning it takes more time for the capacitor to charge or discharge. The capacitance largely influences the circuit's frequency response, determining how the circuit behaves over different signal frequencies when connected to an AC generator.

Resistance

Resistance, represented by \( R \), is the opposition that a material offers to the flow of electric current. It is measured in ohms (\( \Omega \)). In an RC circuit, the resistor controls the rate of discharge of the capacitor. The resistance component is pivotal because it directly impacts the time constant of the circuit along with the capacitance \( C \). According to the formula \( \tau = R \cdot C \), any change in resistance will affect the time taken by the circuit to reach a particular voltage during charging or discharging. In an AC circuit, resistance is part of the overall impedance, affecting how easily alternating current can flow through the circuit without causing power dissipation.

AC Generator

An AC generator is a device that converts mechanical energy into electrical energy in the form of alternating current (AC). In an RC circuit, the AC generator is vital as it supplies the alternating voltage, causing the circuit to respond differently compared to a DC supply. The AC generator's frequency \( f \) significantly affects the circuit's behavior, particularly the impedance \( Z \). As frequency fluctuates, so does the capacitive reactance, consequently altering the impedance. This variability allows RC circuits to be used in diverse applications such as filtering and signal processing, where frequencies need to be selectively amplified or attenuated. Understanding the role of an AC generator in an RC circuit is fundamental for analyzing how the circuit will react under different operational conditions.