Two metal spheres `("radii" r_1, r_2 "with" r_1 lt r_2)` are very far apart but are connected by a thin wire. If their combined charges is Q, then what is their common potential ?

Options

A. `k Q//(r_1 + r_2)`
B. `kQ//(r_1 - r_2)`
C. `-k Q//(r_1 + r_2)`
D. `-kQ//r_1 r_2`

Explanation

To solve the problem, we need to find the common potential of two metal spheres connected by a thin wire, given their combined charge \( Q \) and their radii \( r_1 \) and \( r_2 \) (where \( r_1 < r_2 \)).

### Step-by-Step Solution:

1. **Understanding the System**:
We have two metal spheres with radii \( r_1 \) and \( r_2 \) that are far apart and connected by a thin wire. When connected, they will have the same electric potential.

2. **Define the Common Potential**:
Let the common potential of the two spheres be \( V_c \). Since the spheres are conductors and connected by a wire, the potential on both spheres must be equal:
\[ V_1 = V_2 = V_c \]

3. **Using the Formula for Electric Potential**:
The electric potential \( V \) due to a charge \( Q \) at a distance \( r \) is given by:
\[ V = \frac{kQ}{r} \]
where \( k \) is Coulomb's constant.

4. **Setting Up the Equations**:
For sphere 1 with charge \( Q_1 \) and radius \( r_1 \):
\[ V_c = \frac{kQ_1}{r_1} \]
For sphere 2 with charge \( Q_2 \) and radius \( r_2 \):
\[ V_c = \frac{kQ_2}{r_2} \]

5. **Equating the Potentials**:
Since \( V_1 = V_2 \), we can set the two equations equal to each other:
\[ \frac{kQ_1}{r_1} = \frac{kQ_2}{r_2} \]

6. **Eliminating \( k \)**:
We can cancel \( k \) from both sides:
\[ \frac{Q_1}{r_1} = \frac{Q_2}{r_2} \]

7. **Expressing Charges in Terms of Total Charge**:
We know that the total charge \( Q \) is the sum of the charges on both spheres:
\[ Q = Q_1 + Q_2 \]
From the previous equation, we can express \( Q_1 \) and \( Q_2 \) in terms of \( Q \):
\[ Q_1 = \frac{Qr_1}{r_1 + r_2} \]
\[ Q_2 = \frac{Qr_2}{r_1 + r_2} \]

8. **Finding the Common Potential**:
Now substituting \( Q_1 \) into the expression for \( V_c \):
\[ V_c = \frac{kQ_1}{r_1} = \frac{k \left( \frac{Qr_1}{r_1 + r_2} \right)}{r_1} = \frac{kQ}{r_1 + r_2} \]

9. **Final Result**:
Thus, the common potential \( V_c \) of the two spheres is:
\[ V_c = \frac{kQ}{r_1 + r_2} \]

### Conclusion:
The common potential of the two spheres is given by:
\[ \boxed{\frac{kQ}{r_1 + r_2}} \]