Problem 39 - University Physics with Modern Physics

Question

An \(L \cdot R-C\) series circuit has \(L=0.450 \mathrm{H}, C=2.50 \times \)10^{-5} \mathrm{F},\( and resistance\)R . (a) What is the angular frequency of the circuit when \(R=0 ?\) (b) What value must \(R\) have to give a 5.0\(\%\) decrease in angular frequency compared to the value calculated in part (a)?

Step-by-Step Solution

Answer

(a) \(\omega_0 \approx 298 \text{ rad/s}\); (b) \(R \approx 84.03 \text{ Ohms}\).

Step 1: Understanding the Problem

We are given an L-R-C series circuit with values for the inductance \(L\), capacitance \(C\), and need to calculate the angular frequency of the circuit for two different conditions: when \(R = 0\) and when \(R\) causes a reduction in angular frequency by 5%. We will first find the angular frequency when \(R = 0\).

Step 2: Calculating the Angular Frequency for R=0

The formula for the angular frequency of an L-C circuit (when resistance is zero) is given by:\[\omega_0 = \frac{1}{\sqrt{LC}}\]Plug in the given values \(L = 0.450 \, \text{H}\) and \(C = 2.50 \times 10^{-5} \, \text{F}\) into the formula:\[\omega_0 = \frac{1}{\sqrt{0.450 \times 2.50 \times 10^{-5}}}\]Calculate \(\omega_0\).

Step 3: Solving for the Value of \(\omega_0\)

Calculating the expression from Step 2:\[\omega_0 = \frac{1}{\sqrt{0.450 \times 2.50 \times 10^{-5}}} = \frac{1}{\sqrt{1.125 \times 10^{-5}}}\]\[ = \frac{1}{3.354 \times 10^{-3}} \approx 298 \, \text{rad/s}\]This is the angular frequency when \(R = 0\).

Step 4: Determining the Decreased Angular Frequency

For a 5% decrease in angular frequency, the new frequency \(\omega\) is given by:\[\omega = 0.95 \cdot \omega_0 = 0.95 \times 298 \, \text{rad/s} = 283.1 \, \text{rad/s}\]

Step 5: Finding the Required Resistance for 5% Decrease

The angular frequency for a damped RLC circuit is given by:\[\omega = \sqrt{\frac{1}{LC} - \left(\frac{R}{2L}\right)^2}\]Set \(\omega = 283.1\) and solve for \(R\):\[283.1 = \sqrt{\frac{1}{0.450 \times 2.50 \times 10^{-5}} - \left(\frac{R}{2 \times 0.450}\right)^2}\]Square both sides and simplify, then solve the resulting equation for \(R\).

Step 6: Solving for R

Expanding:\[283.1^2 = \frac{1}{1.125 \times 10^{-5}} - \left(\frac{R}{0.900}\right)^2\]\[\approx 80168.61 = 88888.89 - \frac{R^2}{0.81}\]Solving for \(R^2\):\[\frac{R^2}{0.81} = 88888.89 - 80168.61 = 8720.28\]\[R^2 = 8720.28 \times 0.81\]\[R \approx \sqrt{7065.43} \approx 84.03 \, \text{Ohms}\]

Step 7: Final Calculation Verification

Verify the values and ensure the calculations are correct. They fit the condition for a 5% decrease in angular frequency when utilizing this calculated resistance.

Key Concepts

Angular Frequency

Angular frequency, often denoted by \( \omega \), is a measure of how quickly something is oscillating in terms of angles per unit time, usually in radians per second. In the context of an RLC circuit, it tells us how fast the circuit oscillates when energy transitions between the inductor and the capacitor.
In a situation where resistance \( R \) is zero, the angular frequency \( \omega_0 \) can be calculated using the formula:

• \( \omega_0 = \frac{1}{\sqrt{LC}} \)

When you plug in the specific values of inductance (\( L = 0.450 \) H) and capacitance (\( C = 2.50 \times 10^{-5} \) F) given in the problem, you'll find the angular frequency \( \omega_0 \) by solving:

• \( \omega_0 = \frac{1}{\sqrt{0.450 \times 2.50 \times 10^{-5}}} \approx 298 \, \text{rad/s} \)

This helps you understand how fast the oscillations occur in a perfectly balanced L-C circuit without any resistance.

Damped Oscillations

Damped oscillations occur when the energy in an oscillating system decreases over time. In an RLC series circuit, resistance causes these oscillations to gradually lose energy, becoming less and less significant.
When resistance is present, the oscillations become 'damped.' This means that instead of oscillating forever at the same frequency, the system's frequency is reduced and the amplitude of oscillations decreases over time.
To calculate damped angular frequency, \( \omega \), for a circuit, we use the modified formula:

• \( \omega = \sqrt{\frac{1}{LC} - \left(\frac{R}{2L}\right)^2} \)

In our exercise, we are asked to find \( R \) such that \( \omega \) is 5% less than \( \omega_0 \). Therefore, if \( \omega_0 = 298 \, \text{rad/s} \), the damped frequency \( \omega \) we aim for is:

• \( 0.95 \times 298 = 283.1 \, \text{rad/s} \)

This decrease shows how the presence of resistance affects the performance of the circuit, moderating the speed of its oscillations.

Series Circuit

A series circuit is one type of electrical circuit where components are connected along a single path, so the same current flows through all of them. In an L-R-C circuit, this means that the inductor (L), resistor (R), and capacitor (C) are sequentially connected.

• Voltage across each component adds up to the total voltage.
• Current flowing through each component is the same.

In a series RLC circuit, each component affects the overall impedance and hence the behavior of the circuit. Here, the resistive, capacitive, and inductive components define the current's amplitude and phase. This setup directly influences how the system oscillates, reacts to external stimuli, and how quickly it dissipates energy via damping.
Understanding series circuits is fundamental, especially when examining how components interact to affect properties like resonance and damping. Resonance is a vital part of how series circuits accomplish energy transitions and performance optimization.

Inductive Reactance

Inductive reactance, represented by \( X_L \), is the opposition an inductor presents to changes in current. It's similar to resistance, but it only affects the alternating current (AC), contributing to the total impedance in a circuit.
Inductive reactance is calculated using:

• \( X_L = \omega L \)

This means it directly depends on the frequency of the electrical signal \( \omega \) and the inductance \( L \). In our circuit, as we increase the frequency, the inductive reactance also increases, impacting how the circuit would treat different frequencies.
Inductive reactance plays a significant role in RLC circuits, affecting the current's phase relative to the voltage. When the circuit's frequency changes, \( X_L \) alters the impedance, which may lead to circuit resonance or damping conditions. Understanding inductive reactance equips students with insights into how energy conversion and impedance come into play during oscillations in RLC circuits.