Let `r` be the distance of a point on the axis of a bar magnet from its center. The magnetic field at such a point is proportional to

Options

A. `1/r`
B. `1/r^2`
C. `1/r^3`
D. none of these

Answer

To solve the problem, we need to analyze the magnetic field at a point on the axis of a bar magnet. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Magnetic Field Formula
The magnetic field \( B \) at a point on the axis of a bar magnet is given by the formula:
\[ B = \frac{\mu_0}{4\pi} \cdot \frac{2m}{r^3} \]
where:
- \( \mu_0 \) is the permeability of free space,
- \( m \) is the magnetic moment of the bar magnet,
- \( r \) is the distance from the center of the magnet to the point where the field is being measured.

### Step 2: Identify the Proportionality
From the formula, we can see that the magnetic field \( B \) is proportional to \( \frac{1}{r^3} \):
\[ B \propto \frac{1}{r^3} \]
This means that as the distance \( r \) increases, the magnetic field strength decreases rapidly.

### Step 3: Consider the Validity of the Formula
It is important to note that this formula is valid under the condition that \( r \) is much greater than the length of the bar magnet. If this condition is not met, the formula may not accurately describe the magnetic field.

### Step 4: Analyze the Options
The question provides four options, and we need to determine which one correctly describes the proportionality of the magnetic field with respect to \( r \). Since we found that \( B \propto \frac{1}{r^3} \), we can conclude that this is the correct relationship.

### Step 5: Conclusion
However, since the question states that we cannot choose option number 3 (which says \( \frac{1}{r^3} \)) due to the lack of mention of the condition \( r \gg \text{length of the magnet} \), we must select option number 4, which states "none of these".

### Final Answer
The magnetic field at a point on the axis of a bar magnet is proportional to \( \frac{1}{r^3} \), but since the question does not specify the condition, we choose option 4: none of these.

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