Using the loop rule, derive the differential equation for an LCcircuit (Equation$\mathit{L}\frac{{\mathbf{d}}^{\mathbf{2}}\mathbf{q}}{\mathbf{d}{\mathbf{t}}^{\mathbf{2}}}\mathbf{+}\frac{\mathbf{1}}{\mathbf{C}}\mathit{q}\mathbf{=}\mathbf{0}\mathbf{}$).

An LC circuit is given.

Kirchhoff's loop rule states that the sum of all the electric potential differences around a loop is zero. It is also sometimes called Kirchhoff's voltage law or Kirchhoff's second law. By using the loop rule, we can find the differential equation for an LC circuit from equation 30-35, when the emf voltage is given. Further, from the relation between charge and capacitance, by equating this equation, we can find the differential equation.

Formulae:

The voltage equation due to the current rate through an inductor, $\epsilon =-L\frac{di}{dt}\left(\mathrm{i}\right)$

The charge across a capacitor, $q=CV\left(\mathrm{ii}\right)$

Using the loop rule, we can get the voltage equation as follows:

$V-\epsilon =0$

Now, substituting equations (i) and (ii), we can get that

$\frac{q}{c}+L\frac{di}{dt}=0\dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \left(1\right)$

But, the rate of current can be given as:

$\begin{array}{c}\frac{di}{dt}=\frac{d}{dt}\left[\frac{dq}{dt}\right]\\ =\frac{d^{2}q}{d{t}^2}\end{array}$

Now, substituting the above value in equation (1), we can get the required differential equation for the LC circuit as follows:

$\begin{array}{c}\frac{q}{C}+L\frac{d^{2}q}{d{t}^2}=0\\ \frac{d^{2}q}{d{t}^2}+\frac{q}{LC}=0\end{array}$

Hence, the differential equation of a LC circuit is $\frac{d^{2}q}{d{t}^2}+\frac{q}{LC}=0$.