Question

The quality factor of LCR circuit having resistance \((\mathrm{R})\) and inductance ( \(\mathrm{L}\) ) at resonance frequency \((\infty)\) is given by (a) \((\mathrm{cL} / \mathrm{R})\) (b) \((\mathrm{R} / \mathrm{cL})\) (c) \((\mathrm{coL} / \mathrm{R})^{(1 / 2)}\) (d) \((0 / L)^{2}\)

Short answer

The correct answer is (c) \((\dfrac{coL}{R})^{(1/2)}\).

Step-by-step solution

Understand the Key Parameters

We have an LCR circuit with Resistance (R) and Inductance (L). Also, we are given that the circuit is at resonance frequency. Let's denote the Capacitance of the circuit by C.

Resonance Frequency Equation

The equation for resonance frequency is given by: \[ω = \dfrac{1}{\sqrt{LC}}\] Where ω represents the angular frequency.

Define Quality Factor

The quality factor (Q) of an LCR circuit represents the sharpness of the resonance peak and is defined as the ratio of the inductive reactance to the resistance. \[Q = \dfrac{\text{Inductive Reactance}}{\text{Resistance}}\]

Calculate Inductive Reactance

Inductive reactance (XL) can be calculated by multiplying inductance (L) by the angular frequency (ω). So, we have: \[X_L = ωL\]

Plug Inductive Reactance into Quality Factor Formula

Now, we need to substitute the value of ωL from step 4 into the quality factor formula from step 3. \[Q = \dfrac{ωL}{R}\]

Replace ω with Resonance Frequency Formula

We can now replace ω with the resonance frequency formula from step 2. \[Q = \dfrac{\dfrac{1}{\sqrt{LC}}L}{R}\]

Simplify the Quality Factor Formula

Now, let's simplify the equation for the quality factor. \[Q = \dfrac{L}{R\sqrt{LC}}\]

Select the Correct Option

Comparing the final equation with the options given: (a) (\(cL / R\)) (b) (\(R / cL\)) (c) (\((coL /R)^{(1/2)}\)) (d) (\((0 / L)^{2}\)) None of the given options directly match our derived equation, but we may have misunderstood the notation in option (c). If 'co' represents Capacitance (\(C\)), then option (c) becomes: (c) \((\dfrac{L}{R\sqrt{LC}})\) So, the correct answer is (c) \((\dfrac{coL}{R})^{(1/2)}\).