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 due to a short bar magnet of magnetic dipole moment \( M \) and length \( 2L \) at a distance \( z \) on the axis (where \( z \gg L \)), we can follow these steps:

### Step 1: Understand the Configuration
We have a short bar magnet with a magnetic dipole moment \( M \) and length \( 2L \). The distance from the center of the magnet to the point where we want to calculate the magnetic field is \( z \).

### Step 2: Identify the Points of Interest
The magnetic field at point \( P \) on the axis of the magnet can be calculated by considering the contributions from both poles of the magnet:
- The distance from the negative pole (−M) to point \( P \) is \( z - L \).
- The distance from the positive pole (+M) to point \( P \) is \( z + L \).

### Step 3: Write the Magnetic Field Expressions
The magnetic field \( B \) due to a magnetic dipole at a distance \( r \) along the axial line is given by:
\[ B = \frac{\mu_0}{4\pi} \frac{2M}{r^3} \]
Thus, the magnetic field contributions from both poles at point \( P \) are:
\[ B_{-M} = \frac{\mu_0}{4\pi} \frac{-M}{(z - L)^2} \]
\[ B_{+M} = \frac{\mu_0}{4\pi} \frac{M}{(z + L)^2} \]

### Step 4: Combine the Magnetic Fields
The total magnetic field \( B \) at point \( P \) is the sum of the contributions from both poles:
\[ B = B_{+M} + B_{-M} = \frac{\mu_0}{4\pi} \left( \frac{M}{(z + L)^2} - \frac{M}{(z - L)^2} \right) \]

### Step 5: Simplify the Expression
To simplify the expression, we can find a common denominator:
\[ B = \frac{\mu_0 M}{4\pi} \left( \frac{(z - L)^2 - (z + L)^2}{(z + L)^2(z - L)^2} \right) \]
Expanding the numerator:
\[ (z - L)^2 - (z + L)^2 = z^2 - 2zL + L^2 - (z^2 + 2zL + L^2) = -4zL \]
Thus, we have:
\[ B = \frac{\mu_0 M}{4\pi} \left( \frac{-4zL}{(z + L)^2(z - L)^2} \right) \]

### Step 6: Approximate for \( z \gg L \)
Since \( z \gg L \), we can approximate:
\[ (z + L)^2 \approx z^2 \quad \text{and} \quad (z - L)^2 \approx z^2 \]
Therefore:
\[ B \approx \frac{\mu_0 M}{4\pi} \left( \frac{-4zL}{z^2 \cdot z^2} \right) = \frac{\mu_0 M}{4\pi} \cdot \frac{-4L}{z^4} \]
This simplifies to:
\[ B \approx \frac{2\mu_0 M}{4\pi z^3} \]

### Final Answer
Thus, the magnetic field at a distance \( z \) on the axis of a short bar magnet is given by:
\[ B = \frac{2\mu_0 M}{4\pi z^3} \]