Problem 10 - College Physics

Question

\(\cdot \mathrm{A} 65 \Omega\) resistor, an 8.0\(\mu \mathrm{F}\) capacitor, and a 35 \mathrm{mH}\( inductor are connected in series in an ac circuit. Calculate the impedance for a source frequency of (a) 300 \)\mathrm{Hz}\( and (b) 30.0 \)\mathrm{kHz}$ .

Step-by-Step Solution

Answer

(a) 65.02 Ω for 300 Hz, (b) 6597.64 Ω for 30 kHz.

Step 1: Understand the Impedance Formula for a Series RLC Circuit

The impedance \(Z\) of a series RLC (Resistor-Inductor-Capacitor) circuit is given by: \( Z = \sqrt{R^2 + (X_L - X_C)^2} \), where \( R \) is the resistance, \( X_L = 2 \pi f L \) is the inductive reactance, and \( X_C = \frac{1}{2 \pi f C} \) is the capacitive reactance.

Step 2: Calculate Inductive Reactance \(X_L\) and Capacitive Reactance \(X_C\) for 300 Hz

Given \( L = 35 \text{ mH} = 0.035 \text{ H} \), \( C = 8.0 \mu \text{F} = 8.0 \times 10^{-6} \text{ F} \), and \( f = 300 \text{ Hz} \), calculate \( X_L = 2 \pi (300 \text{ Hz})(0.035 \text{ H}) \) and \( X_C = \frac{1}{2 \pi (300 \text{ Hz})(8.0 \times 10^{-6} \text{ F})} \).

Step 3: Calculate \(X_L\) and \(X_C\) Numerically for 300 Hz

\( X_L = 2 \pi \times 300 \times 0.035 = 65.97 \Omega \).\( X_C = \frac{1}{2 \pi \times 300 \times 8.0 \times 10^{-6}} = 66.27 \Omega \).

Step 4: Calculate Impedance \(Z\) for 300 Hz

Substitute the values into the impedance formula: \( Z = \sqrt{(65)^2 + (65.97 - 66.27)^2} = \sqrt{4225 + (-0.3)^2} = \sqrt{4225 + 0.09} = \sqrt{4225.09} \approx 65.02 \Omega \).

Step 5: Calculate Inductive Reactance \(X_L\) and Capacitive Reactance \(X_C\) for 30 kHz

Given \( f = 30.0 \text{ kHz} = 30000 \text{ Hz} \), calculate \( X_L = 2 \pi (30000 \text{ Hz})(0.035 \text{ H}) \) and \( X_C = \frac{1}{2 \pi (30000 \text{ Hz})(8.0 \times 10^{-6} \text{ F})} \).

Step 6: Calculate \(X_L\) and \(X_C\) Numerically for 30 kHz

\( X_L = 2 \pi \times 30000 \times 0.035 = 6597.34 \Omega \). \( X_C = \frac{1}{2 \pi \times 30000 \times 8.0 \times 10^{-6}} = 0.663 \Omega \).

Step 7: Calculate Impedance \(Z\) for 30 kHz

Substitute the values into the impedance formula: \( Z = \sqrt{(65)^2 + (6597.34 - 0.663)^2} = \sqrt{4225 + 43459264.27} = \sqrt{43463489.27} \approx 6597.64 \Omega \).

Key Concepts

Inductive Reactance

When working with alternating current (AC) circuits that include inductors, an important concept to grasp is inductive reactance, symbolized as \(X_L\). Inductive reactance is a measure of the opposition that an inductor presents to changes in current. The formula to calculate the inductive reactance is \(X_L = 2 \pi f L\), where \(f\) is the frequency of the AC signal and \(L\) is the inductance of the inductor.

• The higher the frequency, the greater the inductive reactance. This is because inductors oppose rapid changes in current.
• This characteristic makes inductors useful for filtering signals, blocking high frequencies while allowing direct current (DC) or low-frequency currents to pass.

Understanding how inductive reactance varies with frequency is crucial in tuning circuits and creating filters in electronic systems.

Capacitive Reactance

Capacitors resist changes in voltage, and their opposition to AC is quantified by capacitive reactance, represented as \(X_C\). The formula to determine capacitive reactance is \(X_C = \frac{1}{2 \pi f C}\), where \(f\) is the frequency and \(C\) stands for the capacitance.

• Higher frequencies lead to lower capacitive reactance, meaning capacitors are better conductors of AC at high frequencies.
• This property makes capacitors excellent components for AC applications, such as coupling and decoupling signals or stabilizing voltage levels.

Capacitive reactance is a vital parameter to consider when designing circuits for signal processing, enabling smoother data flow and reducing noise.

Series RLC Circuit

A series RLC circuit is an electrical circuit consisting of a resistor (R), an inductor (L), and a capacitor (C) all connected in series with an AC power source. This type of circuit showcases the interplay between resistance, inductive reactance, and capacitive reactance. In a series RLC circuit:

• The net reactance is the difference between the inductive and capacitive reactance, \(X = X_L - X_C\).
• At certain frequencies, known as the resonant frequency, the inductive and capacitive reactances equalize, thereby minimizing the circuit's reactance.

The series RLC circuit can be used for frequency tuning and filtering since its impedance is strongly frequency-dependent, allowing it to select or reject specific frequencies.

Impedance Formula

The impedance \(Z\) of a series RLC circuit combines resistance and reactance to provide a comprehensive opposition measure to AC. Impedance is given by the formula \(Z = \sqrt{R^2 + (X_L - X_C)^2}\). Here, \(R\) represents the resistance, and \(X_L\) and \(X_C\) are the inductive and capacitive reactances, respectively.

• Impedance takes both magnitudes and phases of voltage and current into account.
• Unlike pure resistive circuits, impedance includes the phase shift introduced by inductors and capacitors.

Comprehending impedance is crucial for designing and analyzing AC circuits, ensuring they function properly under various operational conditions.