Problem 12 - College Physics

Question

A \(1500 \Omega\) resistor is connected in series with a \(350 \mathrm{mH}\) inductor and an ac power supply. At what frequency will this combination have twice the impedance that it has at \(120 \mathrm{~Hz}\) ?

Step-by-Step Solution

Answer

The new frequency at which the impedance is twice its value at 120 Hz is approximately 336 Hz.

Step 1: Understanding Impedance in Series Circuits

In an R-L series circuit, impedance, denoted as \( Z \), is calculated using the formula \( Z = \sqrt{R^2 + (X_L)^2} \), where \( R \) is the resistance and \( X_L \) is the inductive reactance given by \( X_L = 2 \pi f L \).

Step 2: Calculate Initial Impedance at 120 Hz

Given \( R = 1500 \ \Omega \), \( L = 350 \ \text{mH} = 0.35 \ \text{H} \), and \( f = 120 \ \text{Hz} \), first calculate \( X_L = 2 \pi \times 120 \times 0.35 \). Thus, \( X_L \approx 263.89 \ \Omega \). The impedance is then \( Z = \sqrt{1500^2 + 263.89^2} \approx 1523.54 \ \Omega \).

Step 3: Impedance at New Frequency

We seek a frequency \( f' \) such that the impedance is twice the initial impedance \( 2 \times 1523.54 \ \approx 3047.08 \ \Omega \). Using the impedance formula, set up the equation: \( \sqrt{1500^2 + (2\pi f' \times 0.35)^2} = 3047.08 \).

Step 4: Solving for the New Frequency

Square both sides of the equation from Step 3 and rearrange to solve for \( f' \). This gives: \( 1500^2 + (0.7 \pi f')^2 = 3047.08^2 \). Solve to find \( (0.7 \pi f')^2 = 3047.08^2 - 1500^2 \), leading to \( f' \approx 335.89 \ \text{Hz} \) after appropriate calculations.

Key Concepts

Impedance in an R-L Series Circuit

Impedance is a measure of how much a circuit resists the flow of alternating current (AC) at a particular frequency. In an R-L series circuit, which consists of a resistor and an inductor connected in a line, the total impedance, denoted as \( Z \), combines the effects of the resistor and inductor.
To understand impedance, consider the formula:

• \( Z = \sqrt{R^2 + (X_L)^2} \)
• Where \( R \) is resistance and \( X_L \) is inductive reactance.

The resistor has a constant resistance \( R \), but the reactance \( X_L \) changes with frequency, which affects the impedance. The combination of resistance and reactance gives the circuit its total opposition to AC, increasing or decreasing as frequency changes.

Understanding Inductive Reactance

Inductive Reactance is a specific property of inductors in AC circuits that determines how much they resist the change of current. The inductive reactance, denoted as \( X_L \), depends directly on the frequency of the AC flowing through the circuit.
Here’s the formula:

• \( X_L = 2 \pi f L \)
• Where \( f \) is frequency and \( L \) is inductance.

This means that as the frequency \( f \) increases, the reactance \( X_L \) also increases. Essentially, inductors resist changes in current more at higher frequencies, making them crucial in tuning circuits and controlling frequency response.

The Concept of Resonant Frequency

In circuits with both inductors and capacitors, the resonant frequency is a special frequency where the impedance is minimized, resulting in a condition called resonance. Although this specific problem deals with an R-L circuit, understanding resonant frequency helps in broader circuit design.
The resonant frequency \( f_r \) is mostly calculated in RLC circuits, but the principle extends to understanding how frequency affects impedance:

• Resonance occurs when inductive and capacitive reactances are equal and cancel each other out.
• This leads to maximum current flow at resonance if there is little resistance.

Even without capacitance, knowing about resonance aids in comprehending how inductors affect circuit behavior in varying frequency domains.

Exploring the Resistor-Inductor Circuit

A Resistor-Inductor (R-L) circuit is a basic electrical circuit consisting of two components: resistors and inductors. These circuits are fundamental in electronics, serving purposes like filtering signals or controlling circuit responses to AC.
In an R-L series circuit:

• The resistor, with constant resistance \( R \), opposes electrical current flow.
• The inductor, with inductance \( L \), opposes changes in current flow.

This combination affects how voltages and currents phase with each other. In practice, R-L circuits can be found in timing circuits, where a change in frequency alters the impedance, akin to the exercise problem where changing frequency doubled the impedance.