Two chargeed metallic spheres of same size repel each other by a force F. They are now touched with each other and are then separated to same initial distance. Now the force of repulsion is F'. Which of the following are possible?

Options

A. A. `F' = F `
B. B. `F' lt F`
C. C. `F' cancel(=)F`
D. D. `F' lt F `

Answer

To solve the problem, we need to analyze the situation involving two charged metallic spheres that repel each other with a force \( F \). After touching each other and then being separated, we need to determine the new force of repulsion \( F' \) and how it relates to the original force \( F \).

### Step-by-Step Solution:

1. **Understanding the Initial Conditions**:
- Let the charges on the two spheres be \( q_1 \) and \( q_2 \).
- The initial force of repulsion \( F \) between the two spheres can be expressed using Coulomb's Law:
\[ F = k \frac{|q_1 q_2|}{r^2} \]
where \( k \) is Coulomb's constant and \( r \) is the distance between the centers of the spheres.

2. **Touching the Spheres**:
- When the two spheres are touched, the charges redistribute equally because they are identical in size. The new charges on each sphere after touching will be:
\[ q' = \frac{q_1 + q_2}{2} \]
- Therefore, both spheres will have the same charge \( q' \) after they are separated.

3. **Calculating the New Force of Repulsion**:
- After the spheres are separated again to the same distance \( r \), the new force of repulsion \( F' \) can be calculated as:
\[ F' = k \frac{|q' q'|}{r^2} = k \frac{\left(\frac{q_1 + q_2}{2}\right)^2}{r^2} \]
- Expanding this gives:
\[ F' = k \frac{(q_1 + q_2)^2}{4r^2} \]

4. **Comparing \( F' \) with \( F \)**:
- We can express \( F' \) in terms of \( F \):
\[ F' = \frac{(q_1 + q_2)^2}{4|q_1 q_2|} F \]
- Depending on the values of \( q_1 \) and \( q_2 \), we can analyze the relationship between \( F' \) and \( F \):
- If \( q_1 = q_2 \), then \( F' = F \).
- If \( q_1 \neq q_2 \), \( F' \) could be greater than \( F \) or less than \( F \) depending on the specific values of \( q_1 \) and \( q_2 \).

5. **Conclusion**:
- The possible relationships between \( F' \) and \( F \) are:
- \( F' = F \) (if both charges are equal)
- \( F' > F \) (if the charges are not equal)
- \( F' \neq F \) (in general, after touching, the forces will not be equal unless the charges were initially equal).

### Possible Answers:
Based on the analysis:
- Option A: \( F' = F \) (possible if charges are equal)
- Option B: \( F' > F \) (possible if charges are not equal)
- Option C: \( F' \neq F \) (generally true)