Question

In series LCR circuit, the capacitance is changed from $C$ to $2C$. The inductance should be changed from $L$ to ______ to obtain same resonance frequency.

• A. $4L$
• B. $L/2$
• C. $L/4$
• D. $2L$

Answer 1

The correct option is B $L/2$

The phenomenon of resonance is common among systems that have a tendency to oscillate at a particular frequency.This frequency is called the natural frequency of oscillation of the system. If such a system is driven by an energy source, whose frequency is equal to the natural frequency of the system, the amplitude of oscillations becomes large and resonance is said to occur.
At resonance,
$X_L=X_C$
ie, ${\omega }_{r}L=\frac{1}{{\omega }_{r}C}$
or, ${\omega }_r=\frac{1}{\sqrt{LC}}$
or $2\pi v_r=\frac{1}{\sqrt{LC}}$
or $v_r=\frac{1}{2\pi \sqrt{LC}}$
or $LC=$ constant (as $v$ remains same)
$\therefore \frac{L_2}{L_1}=\frac{C_1}{C_2}$
or $\frac{L_2}{L}=\frac{C}{2C}$ or $L_2=\frac{L}{2}$