The magnetic field at a distance d from a short bar magnet in longitudinal and transverse positions are in the ratio.

Explanation

To find the ratio of the magnetic field at a distance \( d \) from a short bar magnet in longitudinal and transverse positions, we can follow these steps:

### Step 1: Write the formula for the magnetic field in the longitudinal position.
The magnetic field \( B_1 \) at a distance \( d \) from the center of a short bar magnet in the longitudinal position is given by the formula:
\[ B_1 = \frac{\mu_0 \cdot 2m}{4 \pi d^3} \]
where \( \mu_0 \) is the permeability of free space, \( m \) is the magnetic moment of the bar magnet, and \( d \) is the distance from the center of the magnet.

### Step 2: Write the formula for the magnetic field in the transverse position.
The magnetic field \( B_2 \) at the same distance \( d \) from the center of the bar magnet in the transverse position is given by the formula:
\[ B_2 = \frac{\mu_0 \cdot m}{4 \pi d^3} \]

### Step 3: Set up the ratio of the two magnetic fields.
To find the ratio \( \frac{B_1}{B_2} \), we substitute the expressions for \( B_1 \) and \( B_2 \):
\[ \frac{B_1}{B_2} = \frac{\frac{\mu_0 \cdot 2m}{4 \pi d^3}}{\frac{\mu_0 \cdot m}{4 \pi d^3}} \]

### Step 4: Simplify the ratio.
When we simplify the ratio, the \( \mu_0 \), \( m \), and \( 4 \pi d^3 \) terms cancel out:
\[ \frac{B_1}{B_2} = \frac{2m}{m} = 2 \]

### Step 5: State the final result.
Thus, the ratio of the magnetic field in the longitudinal position to that in the transverse position is:
\[ \frac{B_1}{B_2} = 2:1 \]