Understanding Magnetic Fields of a Short Bar Magnet

The question asks about the ratio of magnetic fields produced by a short bar magnet at a certain distance 'd' in two specific positions: longitudinal (axial) and transverse (equatorial). Understanding these positions and the formulas for the magnetic field is key to solving this problem.

What are Longitudinal and Transverse Positions?

• Longitudinal Position (Axial Position): This is a point located on the axis of the bar magnet, extended outwards from the poles.
• Transverse Position (Equatorial Position): This is a point located on a line perpendicular to the axis of the bar magnet, passing through its center.

Magnetic Field Formulas for a Short Bar Magnet

For a short bar magnet with magnetic moment M, the magnetic field at a distance d (where d is much greater than the length of the magnet) is given by different formulas for the longitudinal and transverse positions.

• Magnetic field in Longitudinal Position (\(B_{longitudinal}\)): The magnetic field at a distance \(d\) on the axis is given by:

  Formula
  \( B_{longitudinal} = \frac{\mu_0}{4\pi} \frac{2M}{d^3} \)

  
  Here, \( \mu_0 \) is the permeability of free space, \( M \) is the magnetic moment of the magnet, and \( d \) is the distance from the center of the magnet to the point.
• Magnetic field in Transverse Position (\(B_{transverse}\)): The magnetic field at a distance \(d\) on the equatorial line is given by:

  Formula
  \( B_{transverse} = \frac{\mu_0}{4\pi} \frac{M}{d^3} \)

  
  Here, \( \mu_0 \), \( M \), and \( d \) have the same meanings as above.

Calculating the Ratio

We need to find the ratio of the magnetic fields in the longitudinal and transverse positions, i.e., \( \frac{B_{longitudinal}}{B_{transverse}} \).

Using the formulas:

\( \frac{B_{longitudinal}}{B_{transverse}} = \frac{\frac{\mu_0}{4\pi} \frac{2M}{d^3}}{\frac{\mu_0}{4\pi} \frac{M}{d^3}} \)

We can cancel out the common terms \( \frac{\mu_0}{4\pi} \) and \( \frac{M}{d^3} \) from the numerator and denominator.

\( \frac{B_{longitudinal}}{B_{transverse}} = \frac{2}{1} \)

So, the ratio of the magnetic fields at a distance 'd' from a short bar magnet in longitudinal and transverse positions is 2 : 1.

Conclusion

The magnetic field is stronger on the axis (longitudinal position) compared to the equatorial line (transverse position) at the same distance for a short bar magnet, specifically being twice as strong.