Chapter 30: Problem 39
An \(L\)-\(R\)-\(C\) series circuit has \(L = 0.450\) 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)?
Short Answer
Step by step solution
Formula for Angular Frequency without Resistance
Substitute Known Values into Angular Frequency Formula
Calculate the Angular Frequency without Resistance
Calculate the New Angular Frequency with Reduced Value
Use Formula for Angular Frequency with Resistance
Substitute Known Values and Solve for \(R\)
Calculate the Value of \(R\)
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Angular Frequency
For an ideal LRC circuit with no resistance, the angular frequency is given by the formula: \( \omega_0 = \frac{1}{\sqrt{LC}} \).
Here, \( L \) is the inductance and \( C \) is the capacitance.
When there’s zero resistance, it’s also commonly referred to as the natural frequency.
- The calculation of \( \omega_0 \) involves substituting the values of \( L \) and \( C \) into the formula.
- Angular frequency is measured in radians per second ( \( \mathrm{rad/s} \)).
Inductance
It’s determined by the coil's properties and surrounding materials.
In the formula for angular frequency without resistance, it plays a critical role since it directly affects the circuit’s natural oscillation.
- Measured in henrys (H), inductance depends on coil dimensions and turns.
- Higher inductance slows down the rate of change of current.
Capacitance
In an LRC circuit, it impacts both the time for charge/discharge cycles and the natural oscillatory behavior.
- Measured in farads (F), capacitance is crucial in setting frequency responses.
- Larger capacitance allows more energy storage but reduces the circuit's frequency.
Resistance
It converts electrical energy into heat, affecting the angular frequency when it is present.
In our exercise, introducing resistance causes a damped oscillation which results in a reduced angular frequency.
- Measured in ohms (\( \Omega \)), resistance determines the rate at which energy is dissipated in the form of heat.
- The presence of resistance reduces the efficiency of energy transfer.
Damped Circuit
The result is a decrease in the maximum amplitude of oscillations, causing them to gradually die out.
In a damped LRC circuit, resistance modifies the angular frequency and is calculated by: \[ \omega = \sqrt{\frac{1}{LC} - \left(\frac{R}{2L}\right)^2} \].
The damped frequency is lower than the natural frequency due to energy loss.
- Energy is lost as heat due to resistance.
- The presence of resistance leads to an exponential decay in amplitude.