The Ratio of Magnetic Field Due to Small Bar Magnet in the End-On Position to that in Broadside-On Position

The magnetic field due to a small bar magnet depends on its orientation with respect to the point where the field is being measured. Two common orientations are the end-on position and the broadside-on position. In this response, we will discuss the ratio of the magnetic field in these two positions for the same distance from the magnet.

End-On Position:
In the end-on position, the small bar magnet is aligned such that one of its poles (either the north or south) is facing directly towards the point where the magnetic field is being measured. This means that the magnetic field lines are perpendicular to the surface of the magnet.

Broadside-On Position:
In the broadside-on position, the small bar magnet is aligned such that its magnetic axis is perpendicular to the line connecting the magnet's center to the point where the magnetic field is being measured. This means that the magnetic field lines are parallel to the surface of the magnet.

Magnetic Field Strength:
The magnetic field strength at a point due to a small bar magnet can be calculated using the formula:

B = (μ₀ / 4π) * (2m / r³)

Where:
- B is the magnetic field strength
- μ₀ is the permeability of free space (constant)
- m is the magnetic moment of the bar magnet
- r is the distance from the magnet to the point where the field is being measured

Ratio of Magnetic Field:
To find the ratio of the magnetic field in the end-on position to that in the broadside-on position, we can compare the formulas for the magnetic field strength in these two positions.

For the end-on position, the magnetic field strength (B₁) can be calculated as:

B₁ = (μ₀ / 4π) * (2m / rₑ³)

For the broadside-on position, the magnetic field strength (B₂) can be calculated as:

B₂ = (μ₀ / 4π) * (m / r_b³)

Taking the ratio of B₁ to B₂:

B₁ / B₂ = [(μ₀ / 4π) * (2m / rₑ³)] / [(μ₀ / 4π) * (m / r_b³)]

Simplifying the expression:

B₁ / B₂ = (2m / rₑ³) / (m / r_b³)
= (2m / rₑ³) * (r_b³ / m)
= 2 * (r_b / rₑ)³

Therefore, the ratio of the magnetic field in the end-on position to that in the broadside-on position is given by 2 * (r_b / rₑ)³.

Explanation:
The ratio of the magnetic field strengths in the end-on and broadside-on positions depends on the distance from the magnet to the point where the field is being measured. The ratio is directly proportional to the cube of the ratio of the distances (r_b / rₑ) between the magnet and the field measurement point in the broadside-on and end-on positions, respectively.

This implies that as the distance from the magnet increases, the ratio of the magnetic field in the end-on position to that in the broad