Question

A series RLC circuit is driven by a generator at frequency $1000\text{Hz}$. The inductance is $90.0\text{mH}$; capacitance is $0.500\text{μF}$; and the phase constant has a magnitude of ${60.0}^{\circ }$ (Take ${\pi }^2=10$)

• A. Here current leads the voltage in phase
• B. Here voltage leads the current in phase
• C. Resistance of circuit is $\frac{80\pi }{\sqrt{3}}\Omega$
• D. At resonance $\omega =\frac{\sqrt{2}}{3}\times {10}^4\text{rad/sec}$

Answer 1

Answer: (B), (C), (D)

The correct options are
B Here voltage leads the current in phase
C Resistance of circuit is $\frac{80\pi }{\sqrt{3}}\Omega$
D At resonance $\omega =\frac{\sqrt{2}}{3}\times {10}^4\text{rad/sec}$

$\omega L=90\times {10}^{-3}\times 2\pi \times 1000=180\pi$
$\frac{1}{\omega C}=\frac{1}{0.5\times {10}^{-6}\times 2\pi \times 1000}=\frac{1000}{\pi }$
$\Rightarrow$ circuit is inductive $V_L>V_R$
$\Rightarrow$ voltage leads the current
$\tan \varphi =\frac{\omega L-\frac{1}{\omega C}}{R}=\frac{80\pi }{R}$
$R=\frac{80\pi }{\sqrt{3}}\Omega$
at resonance,
$\omega =\frac{1}{\sqrt{LC}}=\frac{1}{\sqrt{90\times {10}^{-3}}\times 0.5\times {10}^{-6}}$
$=\frac{1}{\sqrt{45\times {10}^{-9}}}=\frac{{10}^5}{15\times \sqrt{2}}=\frac{\sqrt{2}}{3}\times {10}^4\text{rad/sec}$