Problem 34 - College Physics

Question

A series RLC circuit has a resistance of \(25 \Omega,\) an inductance of \(0.30 \mathrm{H},\) and a capacitance of \(8.0 \mu \mathrm{F}\). (a) At what frequency should the circuit be driven for the maximum power to be transferred from the driving source? (b) What is the impedance at that frequency?

Step-by-Step Solution

Answer

(a) 102.9 Hz; (b) 25 \(\Omega\).

Step 1: Understand the Resonant Frequency Formula

In a series RLC circuit, the resonant frequency is the frequency at which the circuit will have minimum impedance and maximum power transfer. The formula to find the resonant frequency (\(f_r\)) is given by \[ f_r = \frac{1}{2\pi \sqrt{LC}} \] where \(L\) is the inductance and \(C\) is the capacitance.

Step 2: Convert Units Appropriately

Before using the formula, convert the capacitance from microfarads to farads. Since \(C = 8.0 \mu F\), convert it to farads: \(C = 8.0 \times 10^{-6} F\).

Step 3: Apply the Values to the Resonant Frequency Formula

Substitute the given values into the resonant frequency formula: \[ f_r = \frac{1}{2\pi \sqrt{0.30 \times 8.0 \times 10^{-6}}} \] Calculate to find the resonant frequency.

Step 4: Calculate the Resonant Frequency

Compute the resonant frequency: \[ f_r = \frac{1}{2\pi \sqrt{0.30 \times 8.0 \times 10^{-6}}} \approx 102.897 \text{ Hz} \]

Step 5: Calculate the Impedance at Resonance

At resonance, the impedance of the circuit is equal to the resistance because the reactive parts cancel each other out. The impedance \(Z\) is thus equal to the resistance \(R = 25 \Omega\).

Step 6: Write the Short Answer

For part (a), the resonant frequency is approximately 102.9 Hz. For part (b), the impedance at this frequency is 25 \(\Omega\).

Key Concepts

Resonant Frequency

In any RLC circuit, the resonant frequency is key to understanding its behavior. It's the frequency at which the circuit experiences minimum impedance, which is crucial for maximum power transfer. This is because, at this frequency, the inductive and capacitive reactances exactly cancel each other out. This ensures that only the resistance affects the circuit's impedance. To calculate the resonant frequency () in an RLC circuit, you use the formula:\[f_r = \frac{1}{2\pi \sqrt{LC}}\]Here, \(L\) is inductance and \(C\) is capacitance. The units for \(L\) and \(C\) must be in henries and farads, respectively. This ensures accuracy in the computation.

• Why is Resonant Frequency Important? It defines the point of maximum power transfer because it minimizes impedance.
• How to Achieve it: Precisely tune the frequency of the driving source to match \(f_r\).

This concept is particularly useful in applications like radio receivers, where tuning into a specific channel with precision is crucial.

Impedance

Impedance in an RLC circuit is a measure of how much the circuit resists the flow of alternating current (AC). It combines resistance, inductance, and capacitance into a single measure. However, at the resonant frequency, something special happens: the reactance from the inductor and capacitor cancel each other. At resonance, the impedance simplifies to:\[Z = R\]Where \(Z\) is the impedance and \(R\) is the resistance of the circuit. This means the circuit behaves like a simple resistive circuit at resonance.

• Key Takeaway: The lower the impedance, the easier it is for current to flow through the circuit.
• At Resonance: The impedance is minimized to just the resistive component.

Understanding impedance is critical for ensuring optimal circuit performance, especially in audio and radio frequency applications where clear signal transmission is vital.

Power Transfer

Power transfer in an RLC circuit is maximized at the resonant frequency. This is due to the minimized impedance caused by the balance of inductive and capacitive reactance. The power absorbed by the circuit from the source depends on several factors: the voltage, the current, and the circuit's impedance. With an RLC circuit tuned to its resonant frequency, the formula for power () transfer is given by:\[P = I^2 R\]In this formula, \(I\) is the current through the circuit and \(R\) is the resistance. The power is at its peak value because at resonance, the current is also at its maximum, given the minimized impedance.

• Why Focus on Maximum Power Transfer? Ensures efficient use of energy and optimal performance of electronic devices.
• Key to Maximizing Power: Maintain the circuit at its resonant frequency condition.

This is particularly essential in power systems and antenna designs where efficiency and minimizing energy loss is crucial.