ODE of electric circuit consisting emf source, inductor and capacitor

[songoku]

Homework Statement

Please see below

Relevant Equations

Kirchhoff Voltage Law

Integration

(a)
$$\varepsilon (t)=V_L+V_C$$
$$\varepsilon (t)=L\frac{dI}{dt}+\frac{q(t)}{C}$$
$$\varepsilon (t)=L\frac{d^2 q}{dt}+\frac{q(t)}{C}$$

(b) taking $q(t)=e^{rt}$,
$$0=L\frac{d^2 q}{dt}+\frac{q(t)}{C}$$
$$0=L.r^2 e^{rt}+\frac{e^{rt}}{C}$$
$$0=L.r^2+\frac{1}{C}$$
$$r^2=-\frac{1}{LC}$$

Is it possible for the value of $r$ to be complex number? Thanks

 

[Orodruin]

Yes. It just means the circuit is oscillatory. Since there is no resistance it is also not dissipative and therefore has no real part.

 

[songoku]

Yes. It just means the circuit is oscillatory. Since there is no resistance it is also not dissipative and therefore has no real part.

Ok, so $r=\pm i\sqrt{\frac{1}{LC}}$ and the general solution will be $q(t)=a_1 e^{i\sqrt{\frac{1}{LC}}t}+a_2 e^{-i\sqrt{\frac{1}{LC}}t}$ where $a_1$ and $a_2$ are constant.

Is this correct?

(c) I don't know to how relate $\omega_o$ to the equation in part (a). Is there any part of the equation actually contains $\omega_o$?

Thanks

 

[Orodruin]

Yes it is correct. Appropriately assigning the constants will result in a real solution. You can also rewrite in terms of sines and cosines. The natural frequency should be evident from this.

 

[songoku]

Thank you very much Orodruin