# A LCR circuit behaves like a damped harmonic oscillator.

Question:

A LCR circuit behaves like a damped harmonic oscillator. Comparing it with a physical spring-mass damped oscillator having damping constant ' $b$ ', the correct equivalence would be:

1. (1) $L \leftrightarrow m, C \leftrightarrow k, R \leftrightarrow b$

2. (2) $L \leftrightarrow \frac{1}{b}, C \leftrightarrow \frac{1}{m}, R \leftrightarrow \frac{1}{k}$

3. (3) $L \leftrightarrow k, C \leftrightarrow b, R \leftrightarrow m$

4. (4) $L \leftrightarrow m, C \leftrightarrow \frac{1}{k}, R \leftrightarrow b$

Correct Option: , 4

Solution:

(4) In damped harmonic oscillation,

$\frac{m d^{2} x}{d t^{2}}=-k x-b v$

$\Rightarrow \frac{m d^{2} x}{d t^{2}}+b \frac{d x}{d t}+k x=0$           ...(1)

In LCR circuit, $\frac{-q}{C}-i R-\frac{L d i}{d t}=0$

$L \frac{d^{2}}{d t^{2}}+R \frac{d q}{d t}+\frac{q}{C}=0$ ...(2)

Comparing equations (i) \& (ii)

$L \leftrightarrow m, C \leftrightarrow \frac{1}{k}, R \leftrightarrow b$