Two equally charged, indentical metal spheres `A` and `B` repel each other with a force `F"`. The spheres are kept fixed with a distance `r` between them. A third identical, but uncharged sphere `C` is brought in contact with `A` and The magnitude of the net electric force on `C` is

Options

A. `F`
B. `3F//4`
C. `F//2`
D. `F//4`

Explanation

To solve the problem, let's break it down step by step.

### Step 1: Understand the Initial Setup
We have two identical metal spheres, A and B, which are equally charged. They repel each other with a force \( F \) when they are a distance \( r \) apart.

### Step 2: Determine the Charge on Spheres A and B
Let the charge on each sphere (A and B) be \( Q \). According to Coulomb's law, the force \( F \) between two charges is given by:
\[ F = k \frac{Q^2}{r^2} \]
where \( k \) is Coulomb's constant.

### Step 3: Bring Sphere C in Contact with Sphere A
When the uncharged sphere C is brought into contact with sphere A, charge will redistribute between A and C. Since they are identical spheres, the charge will be equally shared. After contact, the charge on sphere A will be \( \frac{Q}{2} \) and the charge on sphere C will also be \( \frac{Q}{2} \).

### Step 4: Analyze the Forces Acting on Sphere C
Now, we need to find the net electric force acting on sphere C when it is placed between spheres A and B.

1. **Force due to Sphere A on Sphere C**: The charge on sphere A is \( \frac{Q}{2} \) and the charge on sphere C is \( \frac{Q}{2} \). The distance between A and C is \( \frac{r}{2} \). The force \( F_{AC} \) can be calculated using Coulomb's law:
\[ F_{AC} = k \frac{\left(\frac{Q}{2}\right) \left(\frac{Q}{2}\right)}{\left(\frac{r}{2}\right)^2} = k \frac{\frac{Q^2}{4}}{\frac{r^2}{4}} = k \frac{Q^2}{r^2} \]

2. **Force due to Sphere B on Sphere C**: The charge on sphere B remains \( Q \) and the distance between B and C is \( r \). The force \( F_{BC} \) can be calculated as:
\[ F_{BC} = k \frac{Q \left(\frac{Q}{2}\right)}{r^2} = k \frac{Q^2}{2r^2} \]

### Step 5: Determine the Direction of Forces
- The force \( F_{AC} \) acts to the right (repulsive force between A and C).
- The force \( F_{BC} \) acts to the left (repulsive force between B and C).

### Step 6: Calculate the Net Force on Sphere C
The net force \( F_{net} \) on sphere C can be calculated by subtracting the force due to sphere B from the force due to sphere A:
\[ F_{net} = F_{AC} - F_{BC} = k \frac{Q^2}{r^2} - k \frac{Q^2}{2r^2} \]
\[ F_{net} = k \frac{Q^2}{r^2} \left(1 - \frac{1}{2}\right) = k \frac{Q^2}{r^2} \cdot \frac{1}{2} = \frac{1}{2} k \frac{Q^2}{r^2} \]

### Step 7: Relate to the Initial Force F
Since we know that \( F = k \frac{Q^2}{r^2} \), we can express the net force as:
\[ F_{net} = \frac{1}{2} F \]

### Conclusion
The magnitude of the net electric force on sphere C is:
\[ \frac{F}{2} \]