Question

Resonance occurs in a series $L-C-R$ circuit when the frequency of the applied emf is $1000\text{Hz.}$ Then

• A. When $f=900\text{Hz}$, the circuit behaves as a capacitative circuit
• B. The impendence of the circuit is maximum at $f=1000\text{Hz}$
• C. In resonance, the voltage across $L$ and voltage across $C$ differ in phase by ${180}^{\circ }$
• D.
  If the value of capacitance $C$ is doubled, resonance occurs at $f=2000\text{Hz}$

Answer 1

Answer: (A), (C)

In resonance, the voltage across $L$ and voltage across $C$ differ in phase by ${180}^{\circ }$

Given:
Resonating frequency $f_r=1000\text{Hz}$
At resonance $X_L=X_C$
$X_L=\omega L$ and $\frac{1}{\omega C}=X_C$
Impedence $Z=\sqrt{R^2+{(X_L-X_C)}^2}$
$Z_{min}=R$
If $f$ is decreased, then $\omega$ will decrease and hence $X_C$ will increase therefore at $f<f_r$ circuit behaves as a capacitative.
Phase difference between $V_L$ and $V_C$ is always ${180}^{\circ }$ at any frequency

$\therefore$ Resonating frquency is given by
$f_r=\frac{1}{2\pi \sqrt{LC}}=1000\text{Hz}$
If $C$ is doubled,
$f_r^{\prime }=\frac{1}{2\pi \sqrt{L\left(2C\right)}}=\frac{f_r}{\sqrt{2}}=\frac{1000}{\sqrt{2}}\text{Hz}$