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The A.C. circuit shown in figure. Find the frequency $(ω_{0})$ of the AC voltage source so that current through the source will be in same phase as of voltage of source.

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$ω_{0} = \sqrt{\frac{I}{L C} + \frac{R^{2}}{4 L} + \frac{R}{2 L}}$

$ω_{0} = \sqrt{\frac{I}{L C} + \frac{R^{2}}{L^{2}}}$

$ω_{0} = \frac{1}{\sqrt{L C}}$

$ω_{0} = \sqrt{\frac{I}{L C} - \frac{R^{2}}{L^{2}}}$

answer is B.

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Detailed Solution

$\cos ϕ = \frac{R}{\sqrt{R^{2} + {X_{L}}^{2}}} - - - (1)$

$i_{2} = i_{1} \sin ϕ \Rightarrow \frac{V_{0}}{X_{C}} = \frac{V_{0}}{\sqrt{R^{2} + {X_{L}}^{2}}} \frac{X_{L}}{\sqrt{R^{2} + {X_{L}}^{2}}}$

$\frac{1}{X_{C}} = \frac{X_{L}}{R^{2} + {X_{L}}^{2}} \Rightarrow \frac{1}{\frac{1}{ω_{0} C}} = \frac{ω_{0} L}{R^{2} + {(ω_{0} L)}^{2}}$

$ω_{0} = \sqrt{\frac{I}{L C} - \frac{R^{2}}{L^{2}}}$

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