Question

Derive an expression for magnetic field due to a bar magnet at arbitrary point.​

Answer 1

Answer: please View the attachment for understanding.

Explanation:

Answer 2

Given:

A bar magnet

To find:

An expression for the magnetic field due to the bar magnet at an arbitrary point

Solution:

Let M be the magnetic moment of the magnet.

Let O be the arbitrary point at a distance r from the center of the magnet.

Magnetic field due to the magnet at point O has two components:

1. Vertical component = Bv
2. Horizontal components = Bh

Using the formula for the magnetic field due to a bar magnet at an axial and equatorial point respectively, we have:

Bv = μο M cosθ / 4π r³

Bh = μο 2M cosθ / 4π r³

The total resultant magnetic field B at point O = $\sqrt{Bv^2 + Bh^2}$

Substituting the values,

B = $\sqrt{(\mu o M cos\theta / 4\pi r^3)^2 + (\mu o2 M cos\theta / 4\pi r^3)^2 }$

= μοM / 4πr³ ($\sqrt{4cos^2\theta+sin^2\theta}$)

Using the identity sin²∅ + cos²∅ = 1

B = μοM / 4πr³ $\sqrt{3cos^2\theta+1}$)

Hence, the magnetic field due to a bar magnet at an arbitrary point r distance away from the center of the magnet is μοM / 4πr³ √(3cos²θ+1).