A sinusoidal voltage of peak value 283 , V and angular frequency … - Vidyadip

MCQ

A sinusoidal voltage of peak value $283\, V$ and angular frequency $320/s$ is applied to a series $LCR$ circuit. Given that $R\, = 5\,\Omega $ , $L\,= 25\, mH$ and $C\, = 1000\, \mu F$. The total impedance, and phase difference between the voltage across the source and the current will respectively be

A
$10\,\Omega \,\,$ and $\,\,\,{\tan ^{ - 1}}\left( {\frac{5}{3}} \right)$
✓
$7\,\Omega \,\,$ and $45^o$
C
$10\,\Omega \,\,$ and $\,\,\,{\tan ^{ - 1}}\left( {\frac{8}{3}} \right)$
D
$7\,\Omega \,\,$ and $\,\,\,{\tan ^{ - 1}}\left( {\frac{5}{3}} \right)$

✓

Answer

Correct option: B.

$7\,\Omega \,\,$ and $45^o$

b
Given,

$\mathrm{V}_{0}=283 \,\mathrm{volt}, \omega=320,\, \mathrm{R}=5 \,\Omega, \mathrm{L}=25 \,\mathrm{mH}, \mathrm{C} =1000 \,\mu \mathrm{F}$

$x_{L}=\omega L=320 \times 25 \times 10^{-3}=8\, \Omega$

$x_{C}=\frac{1}{\omega C}=\frac{1}{320 \times 1000 \times 10^{-6}}=3.1 \,\Omega$

Total impedance of the circuit:

$z=\sqrt{\mathrm{R}^{2}+\left(\mathrm{X}_{\mathrm{L}}-\mathrm{X}_{\mathrm{C}}\right)^{2}}=\sqrt{25+(4.9)^{2}}=7\, \Omega$

Phase difference between the voltage and current

$\tan \phi=\frac{\mathrm{X}_{\mathrm{L}}-\mathrm{X}_{\mathrm{C}}}{\mathrm{R}}$

$\tan \phi=\frac{4.9}{5} \approx 1 \Rightarrow \phi=45 \%$

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