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 \( z = 0.1 \, m \) 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 on the axial line of a 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 \( 4\pi \times 10^{-7} \, T \cdot m/A \)
- \( M \) is the magnetic moment of the magnet
- \( z \) is the distance from the center of the magnet to the point where the field is being calculated

### Step 2: Substitute the values into the formula
Given:
- \( M = 10 \, Am^2 \)
- \( z = 0.1 \, m \)

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 expression
The \( 4\pi \) terms cancel out:
\[ B = 10^{-7} \cdot \frac{2 \times 10}{(0.1)^3} \]

Calculating \( (0.1)^3 \):
\[ (0.1)^3 = 0.001 = 10^{-3} \]
Thus, we can rewrite the equation:
\[ B = 10^{-7} \cdot \frac{20}{10^{-3}} = 10^{-7} \cdot 20 \cdot 10^{3} \]

### Step 4: Combine the powers of ten
Now, we combine the powers of ten:
\[ B = 20 \cdot 10^{-7 + 3} = 20 \cdot 10^{-4} \]

### Step 5: Convert to standard form
This can be expressed as:
\[ B = 2 \cdot 10^{-3} \, T \]

### Final Answer
Thus, the magnetic field at a distance of \( z = 0.1 \, m \) from the center of the magnet on the axial line is:
\[ B = 2 \times 10^{-3} \, T \]