Two similarly and equally charged identical metal spheres A and B repel each other with a force of `2xx10^(-5)N`. A third identical unchared sphere C is touched with A and then placed at the midpoint between A and B. Find the net electric force on C.

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the Initial Setup
We have two identical metal spheres A and B, both carrying the same charge \( Q \), which repel each other with a force \( F = 2 \times 10^{-5} \, \text{N} \). A third identical uncharged sphere C is touched with sphere A, acquiring half of its charge, and then placed at the midpoint between A and B.

### Step 2: Determine the Charge on Sphere C
When sphere C is touched by sphere A, it acquires half of the charge from A. Therefore, the charge on sphere C after touching A will be:
\[ Q_C = \frac{Q}{2} \]

### Step 3: Calculate the Forces Acting on Sphere C
After touching A, we need to calculate the forces acting on sphere C due to both spheres A and B.

1. **Force on C due to A**:
The distance between A and C is \( \frac{R}{2} \). The force \( F_{CA} \) between A and C can be calculated using Coulomb's Law:
\[ F_{CA} = k \frac{Q_A \cdot Q_C}{d^2} = k \frac{Q \cdot \frac{Q}{2}}{\left(\frac{R}{2}\right)^2} = k \frac{Q^2/2}{R^2/4} = \frac{2kQ^2}{R^2} \]

2. **Force on C due to B**:
The distance between B and C is also \( \frac{R}{2} \). The force \( F_{CB} \) between B and C is:
\[ F_{CB} = k \frac{Q_B \cdot Q_C}{d^2} = k \frac{Q \cdot \frac{Q}{2}}{\left(\frac{R}{2}\right)^2} = k \frac{Q^2/2}{R^2/4} = \frac{2kQ^2}{R^2} \]

### Step 4: Determine the Direction of Forces
- The force \( F_{CA} \) (force on C due to A) will be directed away from A (to the right).
- The force \( F_{CB} \) (force on C due to B) will also be directed away from B (to the left).

### Step 5: Calculate the Net Force on Sphere C
Since both forces \( F_{CA} \) and \( F_{CB} \) act in opposite directions, the net force \( F_{net} \) on C will be:
\[ F_{net} = F_{CB} - F_{CA} = \frac{2kQ^2}{R^2} - \frac{2kQ^2}{R^2} = 0 \]

### Conclusion
The net electric force on sphere C is zero because the forces due to spheres A and B cancel each other out.

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