The insulated spheres of radii `R_(1)` and `R_(2)` having charges `Q_(1)` and `Q_(2)` respectively are connected to each other. There is

Options

A. no change in the energy of the system
B. an increases in the energy of the system
C. always a decrease in the energy of the system
D. a decrease in energy of the system unless `Q_1R_2 = Q_2R_1`

Answer

To solve the problem of two insulated spheres with charges \( Q_1 \) and \( Q_2 \) connected by a wire, we need to follow these steps:

### Step-by-Step Solution

1. **Understand the Setup**: We have two insulated spheres, Sphere A with radius \( R_1 \) and charge \( Q_1 \), and Sphere B with radius \( R_2 \) and charge \( Q_2 \). When connected by a wire, charge will flow between the spheres until they reach the same electric potential.

2. **Define Electric Potential**: The electric potential \( V \) of a charged sphere is given by the formula:
\[ V = k \frac{Q}{R} \]
where \( k \) is Coulomb's constant, \( Q \) is the charge, and \( R \) is the radius of the sphere.

3. **Set Up the Equation for Equal Potentials**: When the spheres are connected, their potentials become equal:
\[ V_A = V_B \]
This can be expressed as:
\[ k \frac{Q_1 - q}{R_1} = k \frac{Q_2 + q}{R_2} \]
where \( q \) is the amount of charge that flows from Sphere A to Sphere B.

4. **Eliminate \( k \)**: Since \( k \) is a constant, we can cancel it from both sides:
\[ \frac{Q_1 - q}{R_1} = \frac{Q_2 + q}{R_2} \]

5. **Cross Multiply**: Rearranging gives:
\[ (Q_1 - q) R_2 = (Q_2 + q) R_1 \]

6. **Distribute and Rearrange**: Expanding both sides:
\[ Q_1 R_2 - q R_2 = Q_2 R_1 + q R_1 \]
Rearranging to isolate \( q \):
\[ Q_1 R_2 - Q_2 R_1 = q(R_1 + R_2) \]

7. **Solve for \( q \)**: Thus, we find the charge that has flowed:
\[ q = \frac{Q_1 R_2 - Q_2 R_1}{R_1 + R_2} \]

8. **Energy Consideration**: When the spheres are connected, the energy of the system changes. The energy decreases unless the condition \( Q_1 R_2 = Q_2 R_1 \) holds true. This condition means that there is no net charge flow, and thus no change in energy.

9. **Conclusion**: Therefore, the correct statement regarding the energy of the system is that there is a decrease in energy unless \( Q_1 R_2 = Q_2 R_1 \).

### Final Answer
The correct option is:
**A decrease in energy of the system unless \( Q_1 R_2 = Q_2 R_1 \)**.