91Ó°ÊÓ

Inductors $\left(L_1,L_2,……\right)$, Capacitors $\left(C_1,C_2,……\right)$, Resistors$\left(R_1,R_2,……\right)$ are connected in a series.

Kirchhoff's loop rule states that the sum of all the electric potential differences around a loop is zero. By using the loop rule and differential equation for damped oscillations in the RLC circuit, we will prove that the behaviour of LC circuit (a) is identical to that of (b).

Formulae:

The emf equation, according to Kirchhoff’s loop rule,${\mathrm{Î\mu }}_{total}=0$ (i)

Applying the loop rule to circuit (a), we get that the total emf of the circuit is given by,

$\begin{array}{rcl}{\mathrm{Î\mu }}_{total}& =& {\mathrm{Î\mu }}_{L1}+{\mathrm{Î\mu }}_{C1}+{\mathrm{Î\mu }}_{L2}+{\mathrm{Î\mu }}_{R1}+{\mathrm{Î\mu }}_{C2}+{\mathrm{Î\mu }}_{R2}+………….\\ & =& 0\end{array}$
Let$j=1,2,3,…..$

Then, the total emf can be written as:

${\mathrm{Î\mu }}_{total}=\underset{j}{∑}\left({\mathrm{Î\mu }}_{Lj}+{\mathrm{Î\mu }}_{Cj}+{\mathrm{Î\mu }}_{Rj}\right)$

By using the differential of the above equation for the LCR circuit, we get that ,

$\frac{d{\mathrm{Î\mu }}_{total}}{dt}=\underset{j}{∑}\left(L_j\frac{di}{dt}+\frac{q}{C_j}+i{R}_j\right)$

$\begin{array}{rcl}\underset{j}{∑}{L}_j& =& L,\\ \underset{j}{∑}{C}_j& =& C,\\ \underset{j}{∑}{R}_j& =& R\\ & & \end{array}$

$L\frac{di}{dt}+\frac{q}{C}+iR=0\text{(}âˆ\mu \text{from equation (i))}$

This is equivalent to the simple LRC circuit (b).

Hence circuit (a) is identical to the circuit (b).