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 step by step, we will follow the reasoning provided in the video transcript.

### Step 1: Understand the initial charges and their configuration
We have two spheres with charges:
- Sphere 1: \( q_1 = +2 \, \text{nC} \)
- Sphere 2: \( q_2 = -8 \, \text{nC} \)

The distance between them is \( d \).

### Step 2: Calculate the initial force between the charges
Using Coulomb's Law, the force \( F \) between two point charges is given by:
\[ F = k \frac{|q_1 q_2|}{d^2} \]
Substituting the values:
\[ F = k \frac{|(2 \times 10^{-9})(-8 \times 10^{-9})|}{d^2} = k \frac{16 \times 10^{-18}}{d^2} \]

### Step 3: Determine the new charges after touching
When the two spheres touch, the total charge is conserved. The total charge \( Q \) is:
\[ Q = q_1 + q_2 = 2 \, \text{nC} - 8 \, \text{nC} = -6 \, \text{nC} \]
After they touch, the charge will redistribute equally since they are identical spheres. Therefore, each sphere will have:
\[ q' = \frac{Q}{2} = \frac{-6 \, \text{nC}}{2} = -3 \, \text{nC} \]

### Step 4: Calculate the new force with the new charges
Now, the new force \( F' \) between the two spheres, which now both have a charge of \( -3 \, \text{nC} \), is given by:
\[ F' = k \frac{|q' q'|}{r^2} = k \frac{|-3 \times 10^{-9} \times -3 \times 10^{-9}|}{r^2} = k \frac{9 \times 10^{-18}}{r^2} \]

### Step 5: Set the forces equal for the same magnitude
We want the magnitudes of the forces to be equal:
\[ F = F' \]
Thus, we have:
\[ k \frac{16 \times 10^{-18}}{d^2} = k \frac{9 \times 10^{-18}}{r^2} \]
Cancelling \( k \) and rearranging gives:
\[ \frac{16}{d^2} = \frac{9}{r^2} \]

### Step 6: Cross-multiply and solve for \( r \)
Cross-multiplying gives:
\[ 16r^2 = 9d^2 \]
Dividing both sides by 16:
\[ r^2 = \frac{9}{16}d^2 \]
Taking the square root:
\[ r = \frac{3}{4}d \]

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