A solenoid with inductance L = 7 mH and active resistance $R = 44 Ω$ is first connected to a source of direct voltage V₀ and then to a source of sinusoidal voltage with effective value V = V₀. At what frequency of the oscillator will the power consumed by the solenoid be $η = 5 .0$ times less than in the former case?

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5KHz

8KHz

3KHz

2KHz

answer is B.

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Detailed Solution

$\begin{array}{l} P_{DC} = \frac{V_{0}^{2}}{R}; and z = \sqrt{R^{2} + X_{L}^{2}}; \Rightarrow I = \frac{V_{0}}{z} \\ P_{AC} = I^{2} R = (\frac{V_{0}^{2}}{R^{2} + ω^{2} L^{2}}) R \\ P_{DC} = {ηP}_{AC} \Rightarrow \frac{V_{0}^{2}}{R} = η \frac{V_{0}^{2} R}{R^{2} + ω^{2} L^{2}} \\ \Rightarrow R^{2} + ω^{2} L^{2} = {ηR}^{2} \Rightarrow ω^{2} L^{2} = R^{2} (η - 1) \\ ω^{2} = \frac{R^{2}}{L^{2} (η - 1)} \Rightarrow f = \frac{R}{2 πL} \sqrt{η - 1} \Rightarrow f = \frac{44}{2 \times \frac{22}{7} \times 7} \sqrt{5 - 1} \Rightarrow f = 2000 = 2 kHz \end{array}$