A resistor of resistance R, a capacitor of capacitance C and an inductor of inductance L are connected in parallel to an AC power source of voltage ε0 sin ωt. The maximum current through the resistance is half of the maximum current through the power source. Then, the value of R is

Engineering

Physics

AC Circuit Analysis

Question

A resistor of resistance R, a capacitor of capacitance C and an inductor of inductance L are connected in parallel to an AC power source of voltage ε₀ sin ωt. The maximum current through the resistance is half of the maximum current through the power source. Then, the value of R is

$\sqrt{3} | \frac{1}{ωC} - ωL |$

$\sqrt{2} | ωC - \frac{1}{ωL} |$

$\frac{\sqrt{3}}{| ωC - \frac{1}{ωL} |}$

$\sqrt{5} | \frac{1}{ωC} - ωL |$

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${(i_{C} - i_{L})}^{2} + i_{R}^{2} = {i_{0}}^{2}$
$i_{R} = \frac{i_{0}}{2}$
${(i_{C} - i_{L})}^{2} = \frac{3}{4} {i_{0}}^{2}$
$(i_{C} - i_{L}) = \frac{\sqrt{3}}{2} i_{0} = \sqrt{3} iR$
$ωC - \frac{ε}{ωL} = \sqrt{3} \frac{ε}{R}$
$R = \frac{\sqrt{3}}{(ωC - \frac{1}{ωL})}$

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