# (a) Show that an ideal inductor does not dissipate power in an ac circuit.(b) The variation of inductive reactance of an inductor with the frequency (f) of the ac source of 100 V and variable frequency is shown in the fig.(i) Calculate the self-inductance of the inductor.(ii) When this inductor is used in series with a capacitor of unknown value and a resistor of at , maximum power dissipation occurs in the circuit. Calculate the capacitance of the capacitor.

a)

for an inductor circuit voltage leads current by an angle $\frac{\pi }{2}$
Let voltage

$V= V_{m}\sin wt$
current

$I= I_{m}\sin \left ( wt-\frac{\pi }{2} \right )$
Average Power

$p= \frac{\int_{0}^{T}V_{m}\sin wt\, I_{m}\sin \left ( wt-\frac{\pi }{2} \right )dt}{\int_{0}^{T}dt}$
$= -\frac{\frac{1}{2}V_{m}I_{m}\int_{0}^{T}\sin 2wt\, dt}{\int_{0}^{T}dt}= 0$

Thus the average power dissipated over a complete cycle is zero.

b)

$\\X_L=\2\pi fL\\L=\frac{X_{L}}{2\pi f}=\frac{40}{2\pi\times200}=0.032H$

Maximum power dissipated at resonance

$\\\nu=\frac{1}{2\pi\sqrt{LC}}\\\\C=\frac{1}{L\times\nu^2\times4\pi^2}\\\\=\frac{1}{0.1\times9\times10^{4}\times4\pi^2}=8.8\mu F$

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