(a) Prove that an ideal capacitor in an ac circuit does not dissipate power. (b) An inductor of 200 mH, capacitor of 400 and a resistor of 10 <img alt="\

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(a) Prove that an ideal capacitor in an ac circuit does not dissipate power.
(b) An inductor of 200 mH, capacitor of 400 $\mu F$ and a resistor of 10 $\Omega$ are connected
in series to ac source of 50 V of variable frequency. Calculate the
(i) angular frequency at which maximum power dissipation occurs in the
circuit and the corresponding value of the effective current, and
(ii) value of Q-factor in the circuit.

Answers (1)

a) for a capacitor circuit current leads voltage 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

$p= \frac{\int_{0}^{T}V_{m}\sin wt\, I_{m}\sin \left ( wt+\frac{\pi }{2} \right )dt}{\int_{0}^{T}dt}$
      $= \frac{\frac{1}{2}V_{m}I_{m}\int_{0}^{T}\sin 2wt\, dt}{\int_{0}^{T}dt}= 0$
b)
i) Maximum power is obtained at resonance.
At resonance
$X_{L}= X_{C}$
$wL= \frac{1}{wC}$
Angular frequency $w= \frac{1}{\sqrt{LC}}$
                             $= \frac{1}{\sqrt{200\times 10^{-3}\times 400\times 10^{-6}}}$
                            $=\frac{10^{3}}{\sqrt{80}}$
                            $=0\cdot 111\times 10^{3}Hz$
                           $=111\, rad/sc$
$I_{m}=\frac{V_{m}}{Z}= \frac{V_{m}}{R}= \frac{50\sqrt{2}}{10}= 5\sqrt{2}\, A$
ii) Q-factor
   $Q= \frac{1}{R}\sqrt{\frac{L}{C}}$
       $= \frac{1}{10}\sqrt{\frac{200\times 10^{-3}}{400\times 10^{-6}}}$
     $= \frac{1}{10}\sqrt{500}$
    $=\sqrt{5}$