A short bar magnet has a length 2l and a magnetic moment `10 Am^(2)`. Find the magnetic field at a distance of `z = 0.1m` from its centre on the axial line. Here , l is negligible as comppared to z

Options

A. `4 xx 10^(-3) T`
B. `1 xx 10^(-3)T`
C. `3 xx 10^(-3) T`
D. `2 xx 10^(-3) T`

Answer

To solve the problem of finding the magnetic field at a distance \( z = 0.1 \, \text{m} \) from the center of a short bar magnet on its axial line, we can follow these steps:

### Step 1: Understand the formula for the magnetic field of a bar magnet
The magnetic field \( B \) at a distance \( z \) from the center of a short bar magnet on its axial line is given by the formula:
\[ B = \frac{\mu_0}{4\pi} \cdot \frac{2m}{z^3} \]
where:
- \( \mu_0 \) is the permeability of free space, approximately \( 4\pi \times 10^{-7} \, \text{T m/A} \),
- \( m \) is the magnetic moment of the magnet,
- \( z \) is the distance from the center of the magnet.

### Step 2: Substitute the known values
Given:
- \( m = 10 \, \text{Am}^2 \)
- \( z = 0.1 \, \text{m} \)

We can substitute these values into the formula:
\[ B = \frac{4\pi \times 10^{-7}}{4\pi} \cdot \frac{2 \times 10}{(0.1)^3} \]

### Step 3: Simplify the equation
The \( 4\pi \) terms cancel out:
\[ B = 10^{-7} \cdot \frac{20}{(0.1)^3} \]

Calculating \( (0.1)^3 \):
\[ (0.1)^3 = 0.001 \]

Now substituting this back into the equation:
\[ B = 10^{-7} \cdot \frac{20}{0.001} \]
\[ B = 10^{-7} \cdot 20000 \]

### Step 4: Calculate the magnetic field
\[ B = 2 \times 10^{-3} \, \text{T} \]

### Step 5: Conclusion
The magnetic field at a distance of \( z = 0.1 \, \text{m} \) from the center of the short bar magnet on the axial line is:
\[ B = 2 \times 10^{-3} \, \text{T} \]