Question

Resonance occurs in a series $LCR$ circuit when the frequency of the applied emf is $1000Hz$.

• A. When frequency $=900Hz$, then the current through the voltage source will be ahead of emf of the source
• B. The impedance of the circuit is minimum at $f=1000Hz$
• C. At only resonance the voltages across $L$ and $C$ differ in phase by ${180}^{\circ }$
• D. If the value of $C$ is doubled, resonance occurs at $f=2000Hz$

Answer 1

Answer: (A), (B)

The correct options are
A When frequency $=900Hz$, then the current through the voltage source will be ahead of emf of the source
B The impedance of the circuit is minimum at $f=1000Hz$

$f_0=\frac{1}{2\pi \sqrt{LC}}=1000Hz$
when $f=900Hz<f_0\Rightarrow \omega <\frac{1}{\sqrt{LC}}\Rightarrow L\omega <\frac{1}{C\omega }\Rightarrow X_L<X_C\Rightarrow$ circuit is capacitive. therefore the current leads the voltage of the source.
total impedance of circuit = $\sqrt{R^2+(X_L-X_C{)}^2}\Rightarrow$ impedance will be minimum when $X_L-X_C=0$ i.e., impedance will be minimum at resonance frequency of $1000Hz$
voltage across L and C is always differ in phase by 180 degree .
resonant frequency when C is doubled = $\frac{1}{2\pi \sqrt{L2C}}=\frac{1}{\sqrt{2}}\frac{1}{2\pi \sqrt{LC}}=\frac{1000}{\sqrt{2}}Hz$