The magnetic field due to short bar magnet of magnetic dipole moment M and length 21, on the axis at a distance z (where `zgtgtI`) from the centre of the magnet is given by formula

Options

A. `(mu_(0)M)/(4piz^(3))T`
B. `(2mu_(0)M)/(4piz^(3))T`
C. `(4mu_(0)M)/(piz^(3))T`
D. `(mu_(0)M)/(2piz^(3))T`

Answer

To find the magnetic field \( B \) due to a short bar magnet with a magnetic dipole moment \( M \) and length \( 2L \) at a point \( P \) located on the axial line at a distance \( z \) from the center of the magnet, we can follow these steps:

### Step 1: Understand the Setup
We have a short bar magnet with a magnetic dipole moment \( M \) and a length of \( 2L \). The point \( P \) is located on the axial line of the magnet at a distance \( z \) from its center.

### Step 2: Magnetic Field Due to a Magnetic Dipole
The magnetic field \( B \) at a point on the axial line of a magnetic dipole can be expressed using the formula:
\[ B = \frac{\mu_0}{4\pi} \cdot \frac{2M}{z^3} \]
This formula is valid for points far away from the magnet, where \( z \) is much greater than the length of the magnet \( L \) (i.e., \( z \gg L \)).

### Step 3: Calculate the Magnetic Field
Using the formula for the magnetic field on the axial line, we can derive the expression:
1. The distance from the center of the magnet to the ends is \( L \).
2. The magnetic field at point \( P \) can be calculated by considering the contributions from both ends of the dipole.
3. The total magnetic field at point \( P \) is given by:
\[ B = \frac{\mu_0}{4\pi} \cdot \frac{2M}{(z^2 - L^2)^{3/2}} \]
However, since \( z \gg L \), we can simplify this to:
\[ B \approx \frac{\mu_0}{4\pi} \cdot \frac{2M}{z^3} \]

### Step 4: Final Expression
Thus, the magnetic field \( B \) at point \( P \) can be approximated as:
\[ B = \frac{\mu_0}{4\pi} \cdot \frac{2M}{z^3} \]