Question

A solenoid with inductance $L=7\text{mH}$ and active resistance $R=44\Omega$ 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 (in $\text{kHz}$) of the oscillator will the power consumed by the solenoid be $\eta =5$ times less than in the former case?

Answer 1

Power consumed in case of $DC$

$P_{DC}=\frac{V_0^2}{R}$

Power consumed in case of $AC$

$P_{AC}=I^{2}R=\frac{V_0^2}{R^2+{\omega }^2{L}^2}\times R$

$\Rightarrow P_{AC}=\frac{V_0^2/R}{1+{\left(\frac{\omega L}{R}\right)}^2}=\frac{P_{DC}}{\eta }$

$\Rightarrow \frac{\omega L}{R}=\sqrt{\eta -1}$

$\Rightarrow \omega =\frac{R}{L}\sqrt{\eta -1}$

$\Rightarrow f=\frac{R}{2\pi L}\sqrt{\eta -1}$

$\Rightarrow f=\frac{R}{2\pi L}\sqrt{5-1}$

$\Rightarrow f=\frac{R}{\pi L}=\frac{44}{\left(22/7\right)\times 7\times {10}^{-3}}$

$\therefore f=2000\text{Hz}=2\text{kHz}$