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

Options

A. `(Q)/(2)`
B. `(Q)/(sqrt(2))`
C. `(Q)/(sqrt(3)`
D. `(Q)/(3)`

Answer

To solve the problem, we need to determine the charge on the capacitor when the energy is stored equally between the electric field and the magnetic field in an oscillating LC circuit.

### Step-by-Step Solution:

1. **Understand the Total Energy in the LC Circuit**:
The total energy \( E \) stored in an LC circuit is given by the formula:
\[ E = \frac{1}{2} \frac{Q^2}{C} \]
where \( Q \) is the maximum charge on the capacitor and \( C \) is the capacitance.

2. **Energy Distribution**:
When the energy is equally distributed between the electric field (capacitor) and the magnetic field (inductor), we have:
\[ \text{Electric Energy} = \text{Magnetic Energy} \]
Therefore, we can write:
\[ \frac{1}{2} \frac{Q^2}{C} = \frac{1}{2} \frac{q^2}{C} + \frac{1}{2} L i^2 \]
where \( q \) is the charge on the capacitor at that instant, \( L \) is the inductance, and \( i \) is the current.

3. **Equating Energies**:
Since the energies are equal, we can set the electric energy equal to the magnetic energy:
\[ \frac{1}{2} \frac{Q^2}{C} = \frac{1}{2} \frac{q^2}{C} + \frac{1}{2} L i^2 \]
From this, we can simplify:
\[ \frac{Q^2}{C} = \frac{q^2}{C} + L i^2 \]

4. **Relating Current and Charge**:
In an LC circuit, the current \( i \) can be expressed in terms of charge \( q \) as:
\[ i = -\frac{dq}{dt} \]
However, at the moment when the energy is equally divided, we can use the relationship:
\[ i^2 = \left(\frac{Q^2 - q^2}{L}\right) \]
This leads us to the conclusion that:
\[ \frac{Q^2}{C} = \frac{q^2}{C} + \frac{(Q^2 - q^2)}{L} \]

5. **Finding Charge \( q \)**:
Since we want the energy to be equally divided, we can set:
\[ \frac{Q^2}{C} = 2 \cdot \frac{q^2}{C} \]
This simplifies to:
\[ Q^2 = 2q^2 \]
Solving for \( q \):
\[ q^2 = \frac{Q^2}{2} \]
Taking the square root gives:
\[ q = \frac{Q}{\sqrt{2}} \]

### Final Answer:
The charge on the capacitor when the energy is stored equally between the electric and magnetic field is:
\[ q = \frac{Q}{\sqrt{2}} \]