Question

In the figure shown below, switch $S$ is closed at $t=0$ with inductor fully charged. Find the charge $\left(q\right)$ on the capacitor as a function of time $\left(t\right)$.

$i_0$ is the maximum value of current in the circuit.

• A. $q=\sqrt{LC}{i}_0\sin \left(\frac{2t}{\sqrt{LC}}\right)$
• B. $q=\sqrt{LC}\sin \left(\frac{t}{\sqrt{LC}}\right)$
• C. $q=\sqrt{LC}{i}_0\sin \left(\frac{t}{\sqrt{LC}}\right)$
• D. $q=2\sqrt{LC}{i}_0\sin \left(\frac{t}{\sqrt{LC}}\right)$

Answer 1

The correct option is C $q=\sqrt{LC}{i}_0\sin \left(\frac{t}{\sqrt{LC}}\right)$

Let $q_0$ is the maximum charge on the capacitor.

According to principle of conservation of energy,

$\frac{1}{2}L{i}_0^2=\frac{q_0^2}{2C}$

$\Rightarrow q_0=\sqrt{LC}{i}_0$

Also, angular frequency,

$\omega =\frac{1}{\sqrt{LC}}$

We know that, the charge $\left(q\right)$ on the capacitor as a function of time $\left(t\right)$ with inductor fully charged at $t=0$, is

$q=q_0\sin \left(\omega t\right)$

$\Rightarrow q=\sqrt{LC}{i}_0\sin \left(\frac{t}{\sqrt{LC}}\right)$

Hence, option $\left(C\right)$ is the correct answer.