Magnet:

A body which attracts Iron, Cobalt, Nickel, like substances and which exhibits directive property is called Magnet.

Magnetic axis and magnetic meridian :

The line joining the poles of a magnet is called magnetic axis and the vertical plane passing through the axis of a freely suspended magnet is called magnetic meridian.

Geometrical Length (L):

The actual length of magnet is called geometric length.

Magnetic length ($\vec{2l}$ ):

The shortest distance between two poles of a magnet along the axis is called magnetic length or effective length. As the poles are not exactly at the ends the magnetic length is always lesser than geometric length of a magnet.

Magnetic length is a vector quantity. Its direction is from south pole to north pole along its axis

Pole Strength (m):

The ability of a pole to attract or repel another pole of a magnet is called pole strength S.I Unit: ampere – meter. Pole strength is a scalar It depends on the area of cross section of the pole.

Magnetic Moment :

Magnetic dipole and magnetic dipole moment (M):

A configuration of two magnetic poles of opposite nature and equal strength separated by a finite distance is called as magnetic dipole. The product of pole strength (either pole) and magnetic length of the magnet is called magnetic dipole moment or simply magnetic moment.

If ‘m’ be the pole strength of each pole and ‘2l’ be the magnetic length, then magnetic moment M is given by :

$\large M = m \times 2l $

In vector from $\large \vec{M} = m \times \vec{2l} $

Magnetic moment is a vector whose direction is along the axis of the magnet from south to north pole.

The S.I. unit of magnetic moment is ampere-meter² (A-m²) its dimensional formula [AL²]

In electricity the isolated charge q is the simplest structure that can exist. If two such charges of opposite sign are placed near each other, they form an electric dipole characterized by an electric dipole moment $\vec{p}$ .

In magnetism isolated magnetic ‘poles’ which should correspond to isolated electric charges do not exist. The simplest magnetic structure is the magnetic dipole, characterised by a magnetic dipole moment $\vec{M}$ . A current loop, a bar magnet and a solenoid of finite length are examples of magnetic dipoles.

**Variation of magnetic moment due to cutting of magnets:**

Consider a bar magnet of length ‘2l’, pole strength ‘m’ and magnetic moment ‘M’

(i) When the bar magnet is cut into ‘n’ equal parts parallel to its length, then

Pole strength of each part = m/n

( ∵ area of cross section becomes (1/n) times of original magnet)

Length of each part = 2l (remains same)

Magnetic moment of each part, M’ = 2l ×( m/n ) = M/n

(ii) When the magnet is cut into ‘n’ equal parts perpendicular to its length then

Pole strength of each part = m (∵ area of cross section remains same)

Length of each part = 2l/n

Magnetic moment of each part, $\large M’ = \frac{2l}{n} \times m = \frac{M}{n} $

(iii) When the magnet is cut into ‘x’ equal parts parallel to its length and ‘y’ equal parts perpendicular to its length, then

Pole strength of each part $\large = \frac{m}{x} $

Length of each part $\large = \frac{2l}{y} $

Magnetic moment of each part, $\large M’ = \frac{2l}{y} \times \frac{m}{x} = \frac{M}{x y}$

**Variation of magnetic moment due to bending of magnets:**

(i) When a bar magnet is bent, its pole strength remains same but magnetic length decreases. Therefore magnetic moment decreases.

(ii) When a thin bar magnet of magnetic moment M is bent in the form of ⊔- shape with the arms of equal length , then

Magnetic moment of Each part = M/3

Net magnetic moment of the combination,

$\large M’ = \frac{M}{3}(-\hat{j}) + \frac{M}{3}\hat{i} + \frac{M}{3}\hat{j}$

M’ = M/3

(iii) When a thin magnetic needle of magnetic moment M is bent at the middle, so that the two equal parts are perpendicular , then

Magnetic moment of each part = M/2

Net magnetic moment of the combination,

$\large M’ = \frac{M}{2}(-\hat{i}) + \frac{M}{2}\hat{j}$

M’ = M/√2

**Resultant Magnetic Moment due to combination of Magnets:**

(i) When two bar magnets of moments M1 and M2 are joined so that their like poles touch each other and their axes are inclined at an angle ‘θ’, then the resultant magnetic moment of the combination ‘ M’ ’ is given by

$\large M’ = \sqrt{M_1^2 + M_2^2 + 2 M_1 M_2 cos\theta}$

(where , θ = angle between the directions of magnetic moments)

(ii) When two bar magnets of moments M1 and M2 are joined so that their unlike poles touch each other and their axes are inclined at an angle ‘ θ ’, then the resultant magnetic moment

$\large M’ = \sqrt{M_1^2 + M_2^2 + 2 M_1 M_2 cos(180-\theta )}$

[∵ angle between directions of magnetic moments is (180° – θ)]

$\large M’ = \sqrt{M_1^2 + M_2^2 – 2 M_1 M_2 cos\theta}$

(iii) When identical magnets each of magnetic moment M are arranged to form a closed polygon like a triangle (or) square with unlike poles at each corner, then resultant magnetic moment, M’ = 0.