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

• A. $\frac{Q}{2}$
• B. $\frac{Q}{\sqrt{3}}$
• C. $\frac{Q}{\sqrt{2}}$
• D. $Q$

Answer 1

Answer: (C)

Given,the maximum charge on the capacitor is $Q$ .
Let $C$ be its capacitance.
As we know that the energy stored in a capacitor is given as,
$U=\frac{Q^2}{2C}...(1)$
Now, let $q$ be the charge on the capacitor when the energy of the capacitor is equally divided in electric and magnetic fields.
Then, we have,
$\frac{U}{2}=\frac{q^2}{2C}\text{,}\Rightarrow q^2=UC...(2)$
So, from equations $\left(1\right)\text{and}\left(2\right)$ , we get,
$\frac{Q^2}{2C}=\frac{q^2}{C}\text{,}\Rightarrow q^2=\frac{Q^2}{2}\Rightarrow q=\frac{Q}{\sqrt{2}}\text{.}$
Hence, the charge which is equally divided between the electric and magnetic fields is $\frac{Q}{\sqrt{2}}$ .