Question

In an oscillating LC circuit, the maximum charge on the capacitor is $Q$. The charge on the capacitor, when the energy is stored equally between the electric and magnetic field is:

Answer 1

Step 1: Given

The maximum charge on the capacitor is $Q$.

Let $C$ be the capacitance of the capacitor.

$L$ be the inductance of the inductor

Step 2: Formula used

We know that for a fully charged capacitor, the total energy of the LC circuit is given by,

$E=\frac{Q^2}{2C}$
If $q$ is the charge on the capacitor at any instant, $I$ is the current passing through the circuit and $L$ is the inductance of the inductor on the given LC circuit, then

$\frac{Q^2}{2C}=\frac{q^2}{2C}+\frac{L{I}^2}{2}$ $\left(1\right)$

Step 3: Apply the given condition

From the question we have, the energy is stored equally between the electric and magnetic fields.

$\therefore \frac{q^2}{2C}=\frac{L{I}^2}{2}$

Upon substituting this in equation $\left(1\right)$ we get,

$\frac{Q^2}{2C}=\frac{q^2}{2C}\times 2$

$\Rightarrow q^2=\frac{Q^2}{2}$

$\Rightarrow q=\frac{Q}{\sqrt{2}}$

Hence, the required charge on the capacitor is $\frac{Q}{\sqrt{2}}$.