Magnetic intensity for an axial point due to a short bar magnet of magnetic moment `M` is given by

Options

A. (a)`(mu_(0))/(4pi)xx(M)/(d^(3))`
B. (b)`(mu_(0))/(4pi)xx(M)/(d^(2))`
C. (c)`(mu_(0))/(2pi)xx(M)/(d^(3))`
D. (d)`(mu_(0))/(2pi)xx(M)/(d^(2))`

Answer

To find the magnetic intensity (also known as magnetic field strength) at an axial point due to a short bar magnet with a magnetic moment \( M \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Formula**: The magnetic intensity \( B \) at an axial point of a short bar magnet is given by the formula:
\[ B = \frac{\mu_0}{4\pi} \cdot \frac{2M}{d^3} \]
where:
- \( B \) is the magnetic intensity,
- \( \mu_0 \) is the permeability of free space (magnetic constant),
- \( M \) is the magnetic moment of the magnet,
- \( d \) is the distance from the center of the magnet to the point where the intensity is being calculated.

2. **Simplify the Formula**: We can simplify the expression for magnetic intensity:
\[ B = \frac{\mu_0 \cdot 2M}{4\pi d^3} \]
This can be further simplified by reducing the fraction:
\[ B = \frac{\mu_0}{2\pi} \cdot \frac{M}{d^3} \]

3. **Identify the Correct Option**: The expression we derived for magnetic intensity is:
\[ B = \frac{\mu_0 M}{2\pi d^3} \]
We need to check which option corresponds to this expression.

4. **Conclusion**: After checking the options, we find that the correct option is:
\[ B = \frac{\mu_0 M}{2\pi d^3} \]

### Final Answer:
The magnetic intensity for an axial point due to a short bar magnet of magnetic moment \( M \) is given by:
\[ B = \frac{\mu_0 M}{2\pi d^3} \]