Two small identical spheres having charges `+10 muC and - 90muC` attract each other with a force of F newton . If they are kept in contant and then separated by the same distance, the new force between them is -

Options

A. `F/6`
B. 16 F
C. `(16F)/9`
D. 9F

Answer

To solve the problem, we need to analyze the situation step by step.

### Step 1: Understand the initial charges and the force between them.
We have two small identical spheres:
- Sphere A has a charge of \( +10 \, \mu C \)
- Sphere B has a charge of \( -90 \, \mu C \)

The force of attraction between them is given as \( F \).

### Step 2: Calculate the initial force using Coulomb's Law.
Coulomb's Law states that the force \( F \) between two point charges is given by:
\[ F = k \frac{|q_1 q_2|}{r^2} \]
where:
- \( k \) is Coulomb's constant (\( 8.99 \times 10^9 \, N m^2/C^2 \)),
- \( q_1 \) and \( q_2 \) are the charges,
- \( r \) is the distance between the charges.

Substituting the values:
\[ F = k \frac{|(+10 \times 10^{-6})(-90 \times 10^{-6})|}{r^2} \]

### Step 3: Determine the total charge when the spheres are in contact.
When the two spheres are brought into contact, they will share their charges equally because they are identical. The total charge is:
\[ Q_{total} = +10 \, \mu C - 90 \, \mu C = -80 \, \mu C \]
Since they are identical, the charge on each sphere after contact will be:
\[ Q_{new} = \frac{Q_{total}}{2} = \frac{-80 \, \mu C}{2} = -40 \, \mu C \]

### Step 4: Calculate the new force between the spheres after they are separated.
Now, both spheres have a charge of \( -40 \, \mu C \). The new force \( F_{new} \) between them when they are separated by the same distance \( r \) is given by:
\[ F_{new} = k \frac{|q_1 q_2|}{r^2} = k \frac{|-40 \times 10^{-6} \times -40 \times 10^{-6}|}{r^2} \]

Since both charges are negative, the force will be repulsive:
\[ F_{new} = k \frac{(40 \times 10^{-6})^2}{r^2} \]

### Step 5: Determine the sign of the new force.
Since both charges are negative, the force will be repulsive, thus:
\[ F_{new} = + \text{(some value)} \]

### Conclusion:
The new force between the spheres after they are separated will be positive, indicating a repulsive force.

### Final Answer:
The new force between them is \( F_{new} \) (positive).

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