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A capacitor with capacitance C and a coil with active resistance R and inductance L are connected in parallel to a source of sinusoidal voltage of frequency $ω$ .
A circuit consists of a capacitor with capacitance C and a coil with active resistance R and inductance L connected in parallel. Find the impedance of the circuit at frequency $ω$ of alternating voltage

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$| Z | = \sqrt{\frac{R^{2}}{ω^{2} R^{2} C^{2} + {(1 - ω^{2} L C)}^{2}}}$

$| Z | = \sqrt{\frac{R^{2} + ω^{2} L^{2}}{{(1 - ω^{2} L C)}^{2}}}$

$| Z | = \sqrt{\frac{R^{2} + 2 ω^{2} L^{2}}{ω^{2} R^{2} C^{2} + {(1 - ω^{2} L C)}^{2}}}$

$| Z | = \sqrt{\frac{R^{2} + ω^{2} L^{2}}{ω^{2} R^{2} C^{2} + {(1 - ω^{2} L C)}^{2}}}$

answer is D.

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Detailed Solution

$Z = \frac{R + j ω L}{(1 - ω^{2} L C) + ω R C j}$
$∴ | Z | = \frac{| R + j ω L |}{|(1 - ω^{2} L C) + ω R C j|}$
$∴ Z = | Z | = \sqrt{\frac{R^{2} + ω^{2} L^{2}}{ω^{2} R^{2} C^{2} + {(1 - ω^{2} L C)}^{2}}}$

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