(a) Show that an ideal inductor does not dissipate power in an ac circuit.

(b) The variation of inductive reactance (X_L )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 10 \Omega at 300 s^{-1}, maximum power dissipation occurs in the circuit. Calculate the capacitance of the capacitor.

 

 

Answers (1)
S safeer

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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