Question

A radio tuner has a resistance of \(1.00 \mu \Omega\), a capacitance of \(25.0 \mathrm{nF}\) and an inductance of \(3.00 \mathrm{mH}\). a) Find the resonant frequency of this tuner. b) Calculate the power in the circuit if a signal at the resonant frequency produces an emf across the antenna of \(V_{\mathrm{rms}}=1.50 \mathrm{mV}\).

Short answer

Answer: The resonant frequency of the radio tuner is approximately 1.82 MHz, and the power in the circuit when a signal at the resonant frequency is applied is approximately 2.25 mW.

Step-by-step solution

Calculate the resonant frequency

The resonant frequency \(f\) of an RLC circuit is given by the formula: \(f = \frac{1}{2\pi\sqrt{LC}}\) Where \(L\) is the inductance and \(C\) is the capacitance. We are given \(L = 3.00\,\text{mH}\) and \(C = 25.0\,\text{nF}\). Plugging these values into the formula, we get: \(f = \frac{1}{2\pi\sqrt{(3.00 \times 10^{-3}\,\text{H})(25.0 \times 10^{-9}\,\text{F})}}\) Calculate the value of \(f\): \(f \approx 1.82\,\text{MHz}\)

Calculate the power in the circuit

Now, we will calculate the power in the circuit. The power \(P\) in an RLC circuit is given by the formula: \(P = \frac{V_{\text{rms}}^2}{R}\) Where \(V_{\text{rms}}\) is the rms voltage and \(R\) is the resistance. We are given \(V_{\text{rms}} = 1.50\,\text{mV}\) and \(R = 1.00\,\mu\Omega\). Plugging these values into the formula, we get: \(P = \frac{(1.50 \times 10^{-3}\,\text{V})^2}{1.00 \times 10^{-6}\,\text{Ohm}}\) Calculate the value of \(P\): \(P \approx 2.25\,\text{mW}\) In conclusion, the resonant frequency of the radio tuner is approximately \(1.82\,\text{MHz}\), and the power in the circuit when a signal at the resonant frequency is applied is approximately \(2.25\,\text{mW}\).

Key concepts

Radio Tuner Resonance

Resonance in radio tuners is a fascinating phenomenon that allows us to pick up our favorite radio stations clearly. It occurs when the natural frequency of an RLC circuit, composed of a resistor (R), inductor (L), and capacitor (C), is matched by the frequency of a received radio signal. Imagine each component in this circuit as part of a dance, with the inductance and capacitance moving to the rhythm of the electric charge.

At the resonant frequency, the inductive and capacitive reactances are equal in magnitude but opposite in phase, effectively canceling each other out. This causes the circuit to exhibit purely resistive behavior at that frequency, allowing maximum signal strength to be transferred to the tuner, making your radio station come in loud and clear.

In practice, you would adjust either the inductor or capacitor until the circuit is 'tuned' to the desired radio station frequency. To find this magical frequency, we use the formula \( f = \frac{1}{2\pi\sqrt{LC}} \), which for our exercise, yielded the resonant frequency of approximately 1.82 MHz.

RLC Circuit Analysis

Analyzing an RLC circuit involves looking at the interplay between resistance, inductance, and capacitance. When a signal flows through such a circuit, the inductor stores energy in a magnetic field, while the capacitor stores energy in an electric field. The resistor then dissipates the energy as heat. This dance of energy storage and dissipation defines how the circuit reacts to different frequencies of an electrical current.

Given in our exercise, the resistance is \(1.00 \mu \Omega\), quite small, indicating that not much energy is lost as heat and the circuit is nearly ideal. The inductance and capacitance values determine at what frequency this circuit will resonate. When we performed our analysis, we used the resonant frequency formula, which incorporates these values to find the point where our circuit 'sings' the best, which is particularly crucial in applications like radio tuners where clear signal reception is paramount.

Power Calculation in RLC Circuit

Power, in the context of an RLC circuit, is the rate at which energy is dissipated by the resistor. It's crucial when considering the effectiveness and efficiency of electronic devices. To calculate power in an RLC circuit, we use the formula \( P = \frac{V_{\text{rms}}^2}{R} \), where \( V_{\text{rms}} \) is the root-mean-square voltage, and \( R \) is the resistance.

In our radio tuner example, when we reach resonance and a signal is applied at that frequency, the power dissipation reaches its peak. Because the resistance is minimal, the power dissipation is also quite low – only 2.25 mW in our case. This low power consumption is a desirable characteristic, especially in battery-powered devices. Understanding how to calculate power in these scenarios is essential for designing energy-efficient circuits that can operate effectively over long periods.