Suppose the frequency of the source in the previous example can be varied. (a) What is the frequency of the source at which resonance occurs? (b) Calculate the impedance, the current, and the power dissipated at the resonant condition

To address your question about resonance in a system, let’s break it down into manageable parts. Resonance is a phenomenon that occurs when a system is driven at its natural frequency, leading to maximum energy transfer. This can be applied in various contexts, such as electrical circuits or mechanical systems. Here, we’ll focus on a simple electrical resonant circuit involving an inductor and a capacitor.

Finding the Resonant Frequency

The resonant frequency (\(f_0\)) for a simple RLC circuit (which consists of a resistor, inductor, and capacitor) can be calculated using the formula:

f₀ = \frac{1}{2\pi \sqrt{LC}}

In this equation:

• L is the inductance in henries (H).
• C is the capacitance in farads (F).

At this frequency, the inductive reactance (\(X_L\)) and capacitive reactance (\(X_C\)) are equal, resulting in maximum current flow through the circuit.

Calculating Impedance at Resonance

At resonance, the impedance (\(Z\)) of the circuit is at its minimum and is purely resistive. The formula for the impedance in a series RLC circuit is:

Z = R

Here, \(R\) is the resistance in ohms (Ω). This means the total impedance is equal to just the resistance since the reactive components cancel each other out.

Determining Current at Resonance

The current (\(I\)) flowing through the circuit can be calculated using Ohm's Law:

I = \frac{V}{Z}

Where \(V\) is the voltage across the circuit. Since we've established that \(Z = R\) at resonance, we can substitute:

I = \frac{V}{R}

Calculating Power Dissipated at Resonance

The power (\(P\)) dissipated in the circuit can be determined using the formula:

P = I^2 R

Substituting for current from earlier, we can express it as:

P = \left(\frac{V}{R}\right)^2 R = \frac{V^2}{R}

This shows that the power dissipated at resonance is directly proportional to the square of the voltage and inversely proportional to the resistance.

Summary

In summary, the frequency at which resonance occurs can be calculated using the resonant frequency formula, and at this frequency, the impedance is purely resistive, allowing for maximum current to flow through the circuit. The current can be calculated using Ohm’s Law, and the power dissipated can be expressed in terms of voltage and resistance. This interplay of frequency, impedance, current, and power is fundamental in understanding the behavior of resonant circuits.