In L-C oscillatios of a circuit , which of the following is true at `t=3T//4` (T=time period of the oscillation). Assume that at t=0, the capacitor is fully charged?

Options

A. Energy stored in then inductor is zero, while in capacitor is maximum
B. Energy in the inductor and capacitor is shared equallly
C. Energy in the inductor is maximum while in the capacitor is zero
D. none of the above

Answer

To solve the problem regarding the state of an LC oscillator at \( t = \frac{3T}{4} \), we will analyze the energy distribution between the capacitor and the inductor throughout the oscillation cycle.

### Step-by-Step Solution:

1. **Understanding the Initial Condition**:
At \( t = 0 \), the capacitor is fully charged. This means that all the energy in the LC circuit is stored in the capacitor. The energy stored in the capacitor is given by:
\[ U_C = \frac{1}{2} C V^2 = U_0 \]
where \( U_0 \) is the total energy of the system.

**Hint**: Remember that the total energy in an LC circuit remains constant and oscillates between the capacitor and the inductor.

2. **Energy Transfer in the LC Circuit**:
As time progresses, the energy oscillates between the capacitor and the inductor. At specific intervals, all the energy will be in one component:
- At \( t = 0 \): All energy is in the capacitor.
- At \( t = \frac{T}{4} \): All energy is in the inductor.
- At \( t = \frac{T}{2} \): All energy is back in the capacitor.
- At \( t = \frac{3T}{4} \): All energy is again in the inductor.

**Hint**: The energy oscillates between the capacitor and inductor in a periodic manner, with each quarter period representing a complete transfer of energy.

3. **Analyzing the State at \( t = \frac{3T}{4} \)**:
At \( t = \frac{3T}{4} \), the energy will be completely transferred to the inductor. Therefore, the potential energy stored in the capacitor will be zero, and the energy stored in the inductor will be at its maximum:
\[ U_L = U_0 \quad \text{and} \quad U_C = 0 \]

**Hint**: At \( t = \frac{3T}{4} \), think about where the energy has moved. It will be entirely in the inductor.

4. **Conclusion**:
Therefore, at \( t = \frac{3T}{4} \):
- The energy stored in the inductor is maximum, \( U_L = U_0 \).
- The energy stored in the capacitor is zero, \( U_C = 0 \).

The correct statement is that the energy in the inductor is at its maximum at \( t = \frac{3T}{4} \).

### Final Answer:
At \( t = \frac{3T}{4} \), the energy stored in the inductor is maximum, and the energy stored in the capacitor is zero.