A solenoid with inductance $L = 7 mH$ and active resistance $R = 44 Ω$ is first connected to a source of direct voltage $V_{0}$ and then to a source of sinusoidal voltage with effective value $V = V_{0}$ . At what frequency of the oscillator will the power consumed by the solenoid by $η = 5.0$ times less than in the former case?

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10 kHz

4kHz

2 kHz

15 kHz

answer is C.

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Detailed Solution

In the case of direct current,
$P_{D Q} = \frac{V_{M}^{2}}{R}$
AC Circuits
The impedance of LR circuit is

$\begin{array}{l} Z = \sqrt{R^{2} + X_{L}^{2}} = \sqrt{R^{2} + ω^{2} L^{2}} \\ I_{M} = \frac{V_{M}}{Z} = \frac{V_{M}}{\sqrt{R^{2} + ω^{2} L^{2}}} \\ P_{A o} = I_{M}^{2} R = \frac{V_{M}^{2} R}{R^{2} + ω^{2} L^{2}} \end{array}$

According to problem,
$\begin{array}{l} P_{D C} = η P_{A C} \\ \frac{V_{M}^{2}}{R} = η \frac{V_{M}^{2} R}{R^{2} \{1 + {(\frac{ω L}{R})}^{2}\}} \end{array}$

$\begin{array}{l} \frac{V_{If}^{2}}{R} = \frac{η V_{M}^{2}}{R \{1 + {(\frac{ω L}{R})}^{2}\}} \\ \frac{1}{R} = \frac{η}{R \{1 + {(\frac{ω L}{R})}^{2}\}} \\ \{1 + {(\frac{ω L}{R})}^{2}\} = η; or \frac{ω L}{R} = \sqrt{η^{2} - 1} \\ ω = \frac{R}{L} \sqrt{η^{2} - 1}; or 2 π v = \frac{R}{L} \sqrt{n^{2} - 1} \\ v = \frac{R}{2 π L} \sqrt{η^{2} - 1} = 2 kHz \end{array}$