A charge `Q_1` exerts some force on a second charge `Q_2`. If a 3rd charge `Q_3` is brought near, the force of `Q_1` exerted on `Q_2` :-

Options

A. will increase
B. will decrease
C. will remain unchanged
D. will increase if `Q_3` is of the same sign as `Q_1` and will decrease if `Q_3` is of opposite sign

Answer

To solve the question regarding the force exerted by charge \( Q_1 \) on charge \( Q_2 \) when a third charge \( Q_3 \) is introduced, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Initial Setup:**
- We have two charges: \( Q_1 \) and \( Q_2 \).
- The force \( F_{12} \) exerted by \( Q_1 \) on \( Q_2 \) can be calculated using Coulomb's Law:
\[ F_{12} = k \frac{|Q_1 Q_2|}{r^2} \]
where \( k \) is Coulomb's constant and \( r \) is the distance between \( Q_1 \) and \( Q_2 \).

2. **Introducing the Third Charge:**
- A third charge \( Q_3 \) is brought near \( Q_2 \). This charge will exert a force on \( Q_2 \) as well.
- The force \( F_{23} \) exerted by \( Q_3 \) on \( Q_2 \) is given by:
\[ F_{23} = k \frac{|Q_2 Q_3|}{x^2} \]
where \( x \) is the distance between \( Q_2 \) and \( Q_3 \).

3. **Analyzing the Forces:**
- The force \( F_{12} \) remains unchanged because it depends only on the charges \( Q_1 \) and \( Q_2 \) and the distance \( r \) between them. The introduction of \( Q_3 \) does not affect \( F_{12} \).
- The net force acting on \( Q_2 \) will be the vector sum of the forces acting on it:
\[ F_{\text{net}} = F_{12} + F_{23} \]
- However, the question specifically asks about the force exerted by \( Q_1 \) on \( Q_2 \), which is still \( F_{12} \).

4. **Conclusion:**
- The force exerted by \( Q_1 \) on \( Q_2 \) remains unchanged despite the presence of the third charge \( Q_3 \). Thus, we conclude:
\[ F_{12} \text{ remains unchanged.} \]