Consider two identical charges placed a distance `2 d` apart on the `x`-axis.The equilibrium of a positive test charge placed at the point `O` midway between them may be
(i) neutral
(ii) stable for displacement along the `x`-axis
(iii) stable for displacement along the `y`-axis
(iv) unstable for displacement along the `x`-axis

Options

A. stable for displacements along the x-axis
B. neutral
C. unstable for displacement along the y-axis.
D. stable for displacements along the y-axis

Explanation

To analyze the equilibrium of a positive test charge placed at point O, which is midway between two identical charges on the x-axis, we can follow these steps:

### Step 1: Understand the Setup
We have two identical positive charges, each denoted as \( Q \), placed a distance of \( 2d \) apart on the x-axis. The test charge \( q_0 \) is placed at point O, which is located at the midpoint between the two charges. Therefore, the distance from point O to each charge is \( d \).

### Step 2: Analyze Forces in the x-direction
When the test charge \( q_0 \) is at point O and is displaced slightly to the left or right along the x-axis:
- If displaced to the left, the distance to the left charge decreases while the distance to the right charge increases.
- The electrostatic force from the left charge (let's call it \( F_L \)) increases because it is inversely proportional to the square of the distance, while the force from the right charge (let's call it \( F_R \)) decreases.

Since \( F_L \) becomes greater than \( F_R \) when displaced to the left, the net force will push the test charge back towards point O. This indicates that the equilibrium is stable for displacements along the x-axis.

### Step 3: Analyze Forces in the y-direction
Now, consider displacing the test charge \( q_0 \) in the y-direction:
- The forces from both charges will have horizontal components that cancel each other out due to symmetry, but they will have vertical components that will add up.
- The net force will act in the direction of the displacement, meaning if the charge is displaced upward, it will continue to move upward, and if displaced downward, it will continue to move downward.

This indicates that the equilibrium is unstable for displacements along the y-axis.

### Step 4: Conclusion
Based on the analysis:
- The equilibrium is stable for displacement along the x-axis.
- The equilibrium is unstable for displacement along the y-axis.

Thus, the correct answer is that the equilibrium of the positive test charge placed at point O is:
- (ii) stable for displacement along the x-axis
- (iv) unstable for displacement along the y-axis

### Summary of Answers
- (i) Neutral: Incorrect
- (ii) Stable for displacement along the x-axis: Correct
- (iii) Stable for displacement along the y-axis: Incorrect
- (iv) Unstable for displacement along the x-axis: Incorrect