A series LCR circuit containing a resistance 'R' has angular frequency ' $\omega$ '. At resonance the…

A series LCR circuit containing a resistance 'R' has angular frequency ' $\omega$ '. At resonance the voltage across resistance and inductor are ' $\mathrm{V}_{\mathrm{R}}$ ' and ' $\mathrm{V}_{\mathrm{L}}$ ' respectively, then value of inductance 'L' will be
  1. $\frac{V_R R}{V_L \omega}$
  2. $\frac{V_L}{V_R R \omega}$
  3. $\frac{V_R \omega}{V_L R}$
  4. $\frac{\mathrm{V}_{\mathrm{L}} \mathrm{R}}{\mathrm{V}_{\mathrm{R}}(\omega)}$

Solution

At resonance, $Z=R$
Voltage across resistance, $V_R=I \times Z=I \times R ...(i)$
Voltage across inductor, $\mathrm{V}_{\mathrm{L}}=\mathrm{I} \times \mathrm{X}_{\mathrm{L}}=\mathrm{I} \times \omega \mathrm{L}...(ii)$
Dividing (ii) by (i), $\frac{\mathrm{V}_{\mathrm{L}}}{\mathrm{~V}_{\mathrm{R}}}=\frac{\omega \mathrm{L}}{\mathrm{R}} \Rightarrow \mathrm{~L}=\frac{\mathrm{V}_{\mathrm{L}} \mathrm{R}}{\mathrm{~V}_{\mathrm{R}} \omega}$

Asked in: MHT CET 2024 (03 May Shift 2)

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