Magnitude of Equatorial and Axial Fields due to a Bar Magnet

To determine the magnitude of the equatorial and axial fields due to a bar magnet, we can use the formula:

B = (μ₀/4π) * (2m/r³)

where B is the magnetic field, μ₀ is the permeability of free space (4π x 10^-7 Tm/A), m is the magnetic moment of the bar magnet, and r is the distance from the midpoint of the bar magnet.

Given:
Length of the bar magnet (l) = 5 cm = 0.05 m
Distance from the midpoint (r) = 50 cm = 0.5 m
Magnetic moment (m) = 0.40 Am²

Magnitude of Equatorial Field:
The equatorial field is the magnetic field at a point located on the equatorial plane of the bar magnet.

Using the formula mentioned above, substituting the given values, we can calculate the magnitude of the equatorial field.

B_equatorial = (μ₀/4π) * (2m/r³)
= (4π x 10^-7 Tm/A) / (4π) * (2 x 0.40 Am²) / (0.5 m)³
= (10^-7 Tm/A) * (0.8 Am²) / (0.125 m³)
= 6.4 x 10^-7 T

Therefore, the magnitude of the equatorial field due to the bar magnet is 6.4 x 10^-7 T.

Magnitude of Axial Field:
The axial field is the magnetic field at a point located on the axis passing through the midpoint of the bar magnet.

Using the same formula as before, but substituting the distance from the midpoint to the point on the axis (r') as 0, we can calculate the magnitude of the axial field.

B_axial = (μ₀/4π) * (2m/r³)
= (4π x 10^-7 Tm/A) / (4π) * (2 x 0.40 Am²) / (0 m)³
= (10^-7 Tm/A) * (0.8 Am²) / (0 m³)
= undefined

As we can see, the magnitude of the axial field is undefined because we cannot divide by zero. This is because the formula assumes a point infinitely close to the bar magnet, which is not physically possible.

In summary, the magnitude of the equatorial field due to the bar magnet is 6.4 x 10^-7 T, while the magnitude of the axial field is undefined at the midpoint of the bar magnet.