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,
$U=\frac{Q^{2}}{2 C}...\left(\right.1\left.\right)$
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,
$\frac{U}{2}=\frac{q^{2}}{2 C}\text{,}\Rightarrow q^{2}=UC...\left(\right.2\left.\right)$
So, from equations $\left(1\right)\text{and}\left(2\right)$ , we get,
$\frac{Q^{2}}{2 C}=\frac{q^{2}}{C}\text{,}\Rightarrow q^{2}=\frac{Q^{2}}{2}\Rightarrow q=\frac{Q}{\sqrt{2}}\text{.}$
Hence, the charge which is equally divided between the electric and magnetic fields is $\frac{Q}{\sqrt{2}}$ .