Two spheres of electric charges `+2nC` and `-8nC` are placed at a distance d apart. If they are allowed to touch each other, whatis the new distance between them to get a repulsive force of same magnitude as before?

Options

A. d
B. `(d)/(2)`
C. `(3d)/(4)`
D. `(4d)/(4)`

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the Initial Situation
We have two spheres with charges:
- Sphere 1: \( Q_1 = +2 \, \text{nC} \)
- Sphere 2: \( Q_2 = -8 \, \text{nC} \)

They are placed at a distance \( d \) apart. The force between them can be calculated using Coulomb's Law.

### Step 2: Calculate the Initial Force (F1)
According to Coulomb's Law, the force \( F_1 \) between two charges is given by:
\[ F_1 = k \frac{|Q_1 \cdot Q_2|}{d^2} \]
Substituting the values:
\[ F_1 = k \frac{|2 \times (-8)|}{d^2} = k \frac{16}{d^2} \]

### Step 3: Determine the Charges After Touching
When the two spheres touch, they will share their charges. The total charge is:
\[ Q_{\text{total}} = Q_1 + Q_2 = 2 \, \text{nC} - 8 \, \text{nC} = -6 \, \text{nC} \]
Since they are identical spheres, the charge on each sphere after touching will be:
\[ Q' = \frac{Q_{\text{total}}}{2} = \frac{-6 \, \text{nC}}{2} = -3 \, \text{nC} \]

### Step 4: Calculate the New Force (F2)
After they are separated again at a new distance \( d_1 \), the force \( F_2 \) will be:
\[ F_2 = k \frac{|Q' \cdot Q'|}{d_1^2} = k \frac{|-3 \cdot -3|}{d_1^2} = k \frac{9}{d_1^2} \]

### Step 5: Set the Forces Equal
To find the new distance \( d_1 \) such that the magnitude of the forces is the same, we set \( F_1 = F_2 \):
\[ k \frac{16}{d^2} = k \frac{9}{d_1^2} \]
We can cancel \( k \) from both sides:
\[ \frac{16}{d^2} = \frac{9}{d_1^2} \]

### Step 6: Cross-Multiply and Solve for \( d_1^2 \)
Cross-multiplying gives:
\[ 16 d_1^2 = 9 d^2 \]
Now, solving for \( d_1^2 \):
\[ d_1^2 = \frac{9}{16} d^2 \]

### Step 7: Take the Square Root
Taking the square root of both sides:
\[ d_1 = \frac{3}{4} d \]

### Final Answer
The new distance \( d_1 \) between the spheres to get a repulsive force of the same magnitude as before is:
\[ d_1 = \frac{3}{4} d \]

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