(III) Determine a formula for the average power P dissipated in an LRC circuit in terms of L, R, C, ω and Vo. (c) Find an approximate formula for the width of the resonance peak in average power, Δω, which is the difference in the two (angular) frequencies where P has half its maximum value. Assume a sharp peak.

Verified step by step guidance

1. Step 1: Start by writing the expression for the voltage across the LRC circuit, which can be represented as $V(t) = V_0 ext{cos}( heta)$. Here, $V_0$ is the peak voltage and $ heta = heta(t)$ is the phase angle.
2. Step 2: Use Ohm's Law to find the current $I(t)$ in the circuit. The impedance $Z$ of the LRC circuit is given by $Z = ext{sqrt}(R^2 + ( ext{omega} L - 1/( ext{omega} C))^2)$, and the current can be calculated using $I(t) = V(t) / Z$.
3. Step 3: Calculate the average power dissipated in the resistor. Power $P$ in terms of the current and resistance is given by $P = I^2 R$. To find the average power over one cycle, use $P_{ ext{avg}} = rac{1}{T} ext{integrate}(I^2(t) R, t, 0, T)$, where $T$ is the period of the cycle.
4. Step 4: To find the width of the resonance peak $ ext{Delta} ext{omega}$, first determine the resonant frequency $ ext{omega}_0 = 1/ ext{sqrt}(LC)$. At resonance, the power is maximum. The width of the resonance peak is defined as the range of frequencies around $ ext{omega}_0$ where the power is at least half of its maximum value.
5. Step 5: Approximate the width of the resonance peak by finding the frequencies $ ext{omega}_1$ and $ ext{omega}_2$ such that the power at these frequencies is half of the maximum power. The width of the resonance peak $ ext{Delta} ext{omega}$ is then $ ext{omega}_2 - ext{omega}_1$. Use the condition $P( ext{omega}_1) = P( ext{omega}_2) = rac{1}{2}P_{ ext{max}}$ to find $ ext{omega}_1$ and $ ext{omega}_2$.

Key Concepts

Average Power in LRC Circuits

In an LRC circuit, the average power dissipated can be expressed in terms of the circuit's resistance (R), inductance (L), capacitance (C), the angular frequency (ω), and the peak voltage (Vo). The formula typically involves the impedance of the circuit, which combines the effects of R, L, and C, and shows how power varies with frequency, particularly at resonance.

Resonance in LRC Circuits

Resonance occurs in an LRC circuit when the inductive reactance equals the capacitive reactance, leading to maximum current and power transfer. At this point, the circuit's impedance is minimized, and the average power reaches its peak value. Understanding resonance is crucial for analyzing how the circuit responds to different frequencies and how it affects power dissipation.

Width of the Resonance Peak (Δω)

The width of the resonance peak, Δω, represents the range of angular frequencies around the resonance frequency where the average power is at least half of its maximum value. This concept is important for characterizing the sharpness of the resonance; a narrower peak indicates a more selective circuit, while a wider peak suggests broader frequency response. The width can be approximated using the quality factor (Q) of the circuit.