Magnetic Substance

 A substance which is affected by a magnetic field is called a magnetic substance. 

There are three types of magnetic substance. 

1) Diamagnetic substances

     Diamagnetic substances are those substances which are repelled by the magnets, e.g., antimony, bismuth, lead, tin, zinc, mercury, gold, phosphorus, etc. 

2) Paramagnetic substances

     Paramagnetic substances are those substances which are attracted by the magnets, e.g., aluminium, platinum, oxygen, manganese, chromium, etc. 

3) Ferromagnetic substances

     Ferromagnetic substances are those substances which are strongly attracted by the magnetics, e.g., iron, cobalt, nickel, etc. 

 

Diamagnetic substances

Substances belonging to this category are repelled by the magnets and have a tendency to move from the region of stronger magnetic field to the weaker one. 

     Electron revolving round the nucleus is equivalent to a current loop. A current loop gives rise to a magnetic moment. In general, magnetic moments due to different electrons point in all sorts of directions, thus, neutralising each other. 

     Let an external magnetic field of strength B (rightward arrow) be applied in a direction perpendicular to the plane of rotation of the electron. As the electron (charge = - e) moves through this field, it experiences a force F (rightward arrow) = - e (v × B) (both v and B is rightward arrow). The electron which revolves in clockwise direction experiences this force along the radius away from the centre [Fig. 1(i)]. While that rotating in anticlockwise direction has this force acting along the radius towards the centre. Net centripetal force in case (i) decreases. This results in a corresponding decrease in linear velocity of electron and hence a consequent decrease of its magnetic moment from M (rightward arrow) to M - ∆M (both are rightward arrow). Opposite in the case for electron revolving in anticlockwise direction [Fig. 1(ii)]. It has a force F = - e (v × B) acting along the radius towards the centre. This increases net centripetal force and a corresponding increase in its magnetic moment from M to M + ∆M (both are rightward arrow). 

Fig. 1.

      So, for one pair of atoms, which has zero resultant magnetic moment in the absence of external field, now, have a resultant magnetic moment given by

     Change in magnetic moment

            = (M + ∆M) - (M - ∆M) = 2∆M (all are rightward arrow). 

     It is clear that this change in magnetic moment is in a direction opposite to the direction of applied field. It is this resultant magnetic moment which accounts for the diamagnetic behaviour of substance. They have a tendency to move from region of stronger field to that of weaker field. 

Properties of diamagnetic substances

1. Diamagnetic substances are repelled by the magnets. They tends to move from a region of stronger magnetic field to the weaker one. 

2. A bar of diamagnetic substance when suspended in a uniform field comes to rest, at right angles to the lines of force. 

3. A diamagnetic substance, in powdered form, placed in a watch glass over two magnetic poles, separated about 10 cm apart, will get collected in the form of a heap in the centre [Fig. 2]. This is due to the reason that magnetic field in the central region is weaker than that in the corner region. 

Fig. 2. Diamagnetic substance is collected in the weaker field region. 

4. Put a diamagnetic substance, in liquid form in a U-tube whose one limb lies in between the two pole pieces. It will be observed that liquid in limb in between pole pieces, gets depressed while that in the other gets raised [Fig. 3].

Fig. 3. Diamagnetic liquid moves from stronger to weaker field region. 

5. Substance is magnetised in a direction opposite to the direction of magnetising field. 

6. Intensity of magnetisation 'I'm is proportional to the strength of magnetising field. 

7. Permeability of diamagnetic substance is less than one (μ < 1).

8. Magnetic induction B in the diamagnetic substances is less than the strength of magnetising field H (B < H). 

9. Susceptibility 'k' of diamagnetic substances is negative. 

10. Susceptibility of diamagnetic substances is independent of temperature. 

Paramagnetic substances

Paramagnetic materials are those which have at least one unpaired electron. The inner field shells of any elements give rise to diamagnetic character because all the electrons are paired in filled shells. However one or more unpaired electrons in the outmost shell have their magnetic moments (M = - e/2  L) remain uncancelled. 

     In the bulk of material these magnetic moments are all arbitrarily distributed in different directions. This can be explained on the fact that :

     "The thermal energies of atomic magnetic dipoles is more than the interaction energies of various dipoles among themselves."

     Thus, in the absence of external applied magnetic field there is no net magnetic moment in the bulk of paramagnetic sample [Fig. 4].

Fig. 4. Paramagnetic sample in the absence of external field. 

     When magnetic field is applied, the individual atomic dipoles try to allign themselves in the direction of field and net magnetic moment is developed in the sample [Fig. 5].

Fig. 5. Paramagnetic sample in external magnetic field. 

     Before we reach this state, when all the magnetic moments have been arranged due to applied field the intensity of magnetisation is found to be :

     (i) proportional to magnetic intensity in the sample i.e., I ∝ H. 

    (ii) inversely proportional to absolute temperature of the sample i.e., I ∝ 1/T

     Combining factors (i) and (ii), I ∝ H/T

or              I = CH/T         or         I/H = C/T

      Here 'C' is the constant of proportionality

     But     I/H = k (magnetic susceptibility) 

                   k = C/T ; this is called Curie's law of magnetism. 

Curie temperature. As the temperature of a ferromagnetic sample is increased, due to thermal energies, the magnetic moments no longer remain aligned inside the ferromagnetic domains. So, the sample loses its magnetism and behaves like a paramagnetic sample. Thus, 

     "The temperature where a ferromagnetic sample becomes paramagnetic is called curie temperature."

     Curie temperature of a new material is given below :

Material	Fe	Ni	Co	Gd	MnAs	MnB
Curie temperature (K)	1043	627	1388	293	318	5778

Properties of paramagnetic substances

1. They are attracted by the magnets. They tends to move from region of weaker magnetic field to the stronger one. 

2. A bar of paramagnetic substance when suspended in a uniform field comes to rest, along the direction of lines of forces of the field. 

3. Place a paramagnetic substance, in powdered form in a watch glass placed on the two poles of a magnet separated about 10 cm apart. The substance gets collected in the form of two steps at the corners as shown in [Fig. 6]. This is due to the reason that magnetic field in the corner reason is stronger than that in the central region. 

Fig. 6. Paramagnetic substance is collected in stronger field region. 

4. Consider some paramagnetic substance in liquid form put a U-tube whose one limb is kept in between the pole pieces of a magnet. The liquid registers a difference of level in the two limbs [Fig. 7]. Level of liquid in the magnetic field is higher than that outside. This indicates that the substance moves from weaker regions of magnetic field to the stronger one. 

Fig. 7. Paramagnetic substance is collected weaker to stronger field. 

5. They are magnetised in the direction of magnetising field. 

6. Intensity of magnetisation 'I' is proportional to 'H'. 

7. Permeability of a paramagnetic substance is slightly greater than one. 

8. Magnetic induction 'B' is greater than 'H'. 

9. Susceptibility 'k' of paramagnetic substance is positive. 

10. Susceptibility 'k' varies inversely as the temperature of substance. 

                  k ∝ I/T

or              k . T = constant. 

     Here 'T' is the temperature of the substance in Kelvin. 

     This relation is known as "Curie Law".

Ferromagnetic substances 

These substances are strongly attracted by the magnets. 

     A ferromagnetic substance, like iron, is composed of tiny crystal like regions, odd in shape and microscopic in size, that fit together as in a mosaic [Fig. 8]. Within each crystal, there are one or more domains and within each domain there is essentially a perfect alignment of the elementary magnetic moments represented by arrow. This arrangement is of random orientation in an unmagnetised sample. 

Fig. 8. Domains in ferromagnetic. 

     When a magnetic field is applied, the atomic dipoles within a domain may suddenly swing around to line up with the external field, or an already aligned domain may grow in size at the expense of an adjacent domain adversely oriented. In other words, a common boundary between two adjacent domains in the same crystal will move. As the field grows stronger, more and more domains flip around suddenly to line up, while others grow in size, until all the domains are aligned. If a ferromagnetic specimen is heated above a certain temperature, called as 'Curie point', the exchange coupling disappears and the substance becomes paramagnetic. 

Properties of ferromagnetic substances

     The properties of ferromagnetic substances are nearly similar to those of paramagnetic substances, but are exhibited on a magnified scale. 

1. They are strongly attracted by the magnets. They tends to move from region of weaker magnetic field to the stronger one. 

2. A bar of ferromagnetic substance, suspended in a magnetic field, comes to rest along the direction of magnetic field. 

3. A ferromagnetic substance, in powdered form gets collected near the magnetic poles as shown in [Fig. 6].

4. A ferromagnetic liquid moves up in the limb of U-tube placed in a magnetic field as shown in [Fig. 7].

5. They are magnetised in the direction of the magnetising field. 

6. Intensity of magnetisation 'I' is not proportional to 'H'. 

7. Permeability of ferromagnetic substance is much greater than one (μ >> 1).

8. Magnetic induction 'B' is much greater than 'H' (B >> H). 

9. Susceptibility of ferromagnetic substances is positive. 

10. Susceptibility of ferromagnetic substances changes with temperature. 'k' decreases with a rise in temperature. Above a certain temperature called Curie point, the substance starts behaving like a paramagnetic substance. On cooling it below Curie point, the ferromagnetic properties are regained. 

 Magnetic Field

Consider an isolated magnetic pole placed at any point. It experiences no force. When another magnetic pole is placed near it, a force starts acting on it. For the first pole, the properties of space around it have undergone a change due to placing of the second pole near it. This space, with modified properties, is called the magnetic field of the second pole. 

Magnetic field, of any magnetic pole, is the region (space) around it in which its magnetic influence can be realised. 

The magnetic field which we plot, in laboratory, is actually a section of magnetic field in the horizontal plane (paper). 

Lines of Force - "Flux Lines"

Imagine a unit magnetic north pole to be situated at any point in a magnetic field. It experiences a force given by Coulomb's law. If the north pole were completely free to move under the action of this force, it would move along a path called lines of force. 

     Lines of force is the path along which a unit north pole would move if it were free to do so. 

     In case of isolated magnetic poles the lines of force is a straight line while in case of a combination of poles (a magnet) it is a curved line. The arrow head on the line indicates the direction of motion of the free north pole. 

Properties of magnetic lines of force

(i) Lines of force are directed away from a north pole and are directed towards a south pole. A line of force starts from a north and ends at a south pole if they are isolated poles. 

(ii) Tangent, at any point, to the magnetic line of force gives the direction of magnetic intensity at the point [Fig. 1].

Fig. 1. Direction of magnetic intensity at a point. 

(iii) Two lines of force never cross each other. If the two lines were to cross, two tangents could be drawn to the line of force at the common point [Fig. 2] meaning thereby two directions of magnetic intensity at that point, which is obviously not possible. 

Fig. 2. Two directions of magnetic intensity at a point. 

(iv) The number of lines of force per unit area (area being perpendicular to lines) is proportional to magnitude of strength of field (magnetic intensity) at that point. Thus, more concentration of lines represents stronger magnetic field. 

(v) The lines of force tends to contract longitudinally or lengthwise i.e., they possess longitudinal strain as shown in [Fig. 3(i)]. Due to this property the two unlike poles attract each other. 

Fig. 3. Magnetic lines of force. 

(vi) The lines of force tends to exert lateral (sideways) pressure, i.e., they repel each other laterally. This explains the repulsion between two similar poles [Fig. 3(ii)].

(vii) 4π lines of force starts from a unit magnetic pole. 

Representation of Magnetic Field

A magnetic field can be represented by a set of magnetic lines of force. If the lines are spaced widely apart, it is a weak field. If the lines are situated close to each other, it represents a strong field. A magnetic field is of two types. 

     (i) Uniform field. A magnetic field is said to be uniform if it has same strength in magnitude and direction at all the points. It represented by a set of parallel lines of force. 

     [Fig. 4(i)] represents a uniform field directed from left to right. [Fig. 4(ii) and (iii)] represents uniform magnetic field at right angles to the plane of the paper directed outwards and inwards respectively. 

Fig. 4. Representation of uniform field. 

     (ii) Non-uniform field. A magnetic field is said to be non-uniform if it has different field strength at different points. It is represented by a set of convergent [Fig. 5(i)] or divergent lines of force [Fig. 5(ii)].

Fig. 5. Non-uniform magnetic field. 

Strength of Magnetic Field : (B) (rightwards arrow) 

The strength of magnetic field, which is a vector quantity, can be defined in any of the following ways:

     (i) In terms of force on a unit north pole

     Strength of magnetic field or magnetic intensity at any point, is defined as the force experienced by a unit north pole placed at that point. The direction of magnetic intensity is the direction in which the unit north pole would move if it were free to do so. 

∴            B = μ₀/4π × m × 1/r² = μ₀/4π × m/r²

Where m is pole strength of source. 

Its unit is NA-1m-1 . 

     (ii) In terms of lines of flux. Magnetic flux density

     We know that a strong field is represented by lines of force crowded together. So, crowding of lines of force can also be used as a measure of field strength. 

     Strength of magnetic field at any point, is defined as the number of flux lines passing through a unit area placed normally to the flux lines at that point. 

           B = magnetic flux/area

     This concept leads to the unit of magnetic flux density as 'Wb m-2' also known as 'tesla' in S.I. and Maxwell cm-2 (known as gauss) in C.G.S. system. 

     (iii) In terms of force on a moving charge. When a charge 'q' moves in a uniform magnetic field of strength B, it experiences a force F (rightwards arrow) , whose magnitude and direction is given by the relation. 

           F = qv B sin θ 

Where 'θ' is the angle which the direction of motion of charge makes with the direction of lines of force of magnetic field. 

∴           B = F/q v sin θ

∴     if q = 1 , v = 1 and sin θ = 1 i.e., θ = 90°

     Thus, strength of magnetic field is numerically equal to the force experienced by a unit charge, moving with a unit velocity at right angles to the direction of lines of force of the field. 

Magnetic Field Intensity Due To A Bar Magnet In Free Space

Case (i) Point situated on the axial line End-on position/Tan A position of gauss

     Consider a point P situated on the axial line, at a distance r from the centre O of the magnet of magnetic length  '2l' [Fig. 6].

Magnetic intensity at Practice due to N-pole only, 

         FN = μ₀/4π × m/NP² along NP produced

or     FN = μ₀/4π × m/(r - l)² along NP produced  

Intensity at Practice due to S-pole, 

         FS = μ₀/4π × m/SP² along SP = μ₀/4π × m/(r + l)² along PS 

If 'F' is the resultant magnetic intensity at Practice due to the magnet, 

     F = FN - FS along NP produced

     F = μ₀/4π [m/(r - l)² - m/(r + l)²] along NP produced

         F = μ₀/4π × m [1/(r - l)² - 1/(r + l)²] along NP produced

or     F = μ₀/4π × m [(r + l)² - (r - l)²/(r² - l²)²] along NP produced

or     F = μ₀/4π × m × 4lr/4π (r² - l²)² along NP produced

or     F = μ₀/4π × 2Mr/(r² - l²)² along NP produced

where M = 2 ml is the magnetic moment of the magnet. 

Fig. 6. Field on the axial line. 

In case of magnetic dipole, the two poles N and S are situated very close to each other. Neglecting l as compared to r. 

          F = μ₀/4π × 2Mr/r⁴ along direction of M

or      F = μ₀/4π × 2M/r³ along direction of M

It is clear that the direction of intensity is along the axial line directed away from the centre if the point is situated on the side of N-pole. 

Case (ii) Point situated on equatorial line Broad side-on position/Tan B position of gauss 

Consider a point P situated, on the equatorial line at a distance r from the centre of a magnet of magnetic length 2l [Fig. 7]. 

Fig. 7. Field on the equatorial line. 

Join NP and SP. 

            NP = SP = √r² + l²

A unit north Pole placed at P (to determine the magnetic intensity at P) experiences a force along PK due to N-pole and along PS due to S-pole. 

Magnetic intensity FN at P due to N-pole only, 

         FN = μ₀/4π × m/(NP)² along PK

or     FN = μ₀/4π × m/r² + l² along PK

Magnetic intensity FS at P due to S-pole only, 

           FS = μ₀/4π × m/SP² along PS

or       FS = μ₀/4π × m/r² + l² along PS

Resolving FN and FS along PX and PY. 

Components of F₁ 

        (i) FN cos θ = μ₀/4π × m/r² + l² cos θ along PX

       (ii) FN sin θ = μ₀/4π × m/r² + l² sin θ along PY

Components of F₂

       (i) FS cos θ = μ₀/4π × m/r² + l² cos θ along PX

      (ii) FS sin θ = μ₀/4π × m/r² + l² sin θ along PO. 

Components FN sin θ along PY and FS sin θ along PO, being equal and opposite cancel each other while FN cos θ and FS cos θ along PX sum up together to give the resultant intensity F at Practice. 

         F = (FN cos θ + FS cos θ) along PX

or     F = [μ₀/4π × m/r² + l² cos θ + μ₀/4π × m/r² + l² cos θ] along PX

or     F = μ₀/4π × 2m/(r² + l²) cos θ along PX

From ΔOSP, cos θ = OS/SP = l/√r² + l²

Substituting for cos θ, we get

         F = μ₀/4π × 2m/(r² + l²) × l/√r² + l² along PX

or     F = μ₀/4π × M/(r² + l²)3/2  along PX

where M = 2 ml is the magnetic moment of the magnet. 

In case of a magnetic dipole, two poles N and S are situated very close to each other. Neglecting l as compared to r, 

∴            F = μ₀/4π × M/r³ along PX

Case (iii) Point situated anywhere

Let People be a point lying at a distance 'r' from the centre of a short magnet NS (magnetic dipole), such that the line joining P with the centre O subtends an angle θ with the axial line [Fig. 8].

Fig. 8. Magnetic field at any point. 

Magnetic moment of the dipole is a vector quantity and is directed along SN. Resolving it into two rectangular components. 

     (i) M cos θ  along OP. 

    (ii) M sin θ perpendicular to OP

Thus, the magnetic dipole can be considered to be equivalent the two magnetic dipoles of moments M cos θ and M sin θ placed perpendicular to each other. 

Point P lies on the axial of dipole of magnetic moment M cos θ. 

Therefore, the magnetic intensity Fax due to this component is

          Fax = μ₀/4π × 2M cos θ/r³ along PY

Point P lies on the equatorial line of dipole of moment M sin θ. 

Therefore, intensity Feq  at P due to this component is

Feq = μ₀/4π × M sin θ/r³ along PX

Since intensities Fax and Feq acting along PY and PX are mutually perpendicular to each other, net intensity F and P is given by

F = √F²ax + F²eq along PZ

or F = μ₀/4π √(2M cos θ/r³)² + (M sin θ/r³)² along PZ

         F = μ₀/4π M/r³ √sin² θ + 4 cos² θ along PZ

F = μ₀/4π M/r³ √sin² θ + cos² θ + 3 cos² θ along PZ

or F = μ₀/4π M/r³ √1 + 3 cos² θ along PZ ... (1)

Direction. In triangle PYZ,

tan β = ZY/PY = M sin θ/r³ × r³/ 2M cos θ

or tan β = 1/2 tan θ ... (2)

Special cases:

(a) If θ = 0, i.e., P lies on axial line.

Using equation (1) and (2), we get

F = μ₀/4π M/r³ √1 + 3 cos² θ along direction of M

F = μ₀/4π × M/r³ along direction of M

and tan β = 1/2 sin θ = 0

i.e., β = 0

The result agrees with that as obtained earlier.

(b) If θ = 90°, i.e., the point lies on equatorial line.

Using equation (1) and (2), we get

F = μ₀/4π M/r³ √1 + 3 cos² 90° along direction opposite to M

F = μ₀/4π M/r³ along direction opposite to M

tan β = 1/2 tan 90° = ∝ i.e., β = 90°

Direction of F will be at right angles to the equatorial line. This result also agrees with that obtained earlier.

Important Notes:

1. Direction of magnetic intensity at any point, on the axial line of a bar magnet is always parallel to the direction of its magnetic moment vector.
2. Direction magnetic intensity at any point, on the equatorial line of a bar magnet is always anti-parallel to the direction of magnetic moment vector.

          

Magnet and Atomic/Molecular theory of Magnetisation

Magnet:

Introduction

Science of magnetism dates back to 600 B.C., when Thales of Miletus had a knowledge that an ore of iron called, magnetite, possessed the properties of attracting small pieces of iron towards it. It had an additional characteristic that, when suspended freely it pointed in a particular direction. That is why it was given a name lodestone or the Leading stone i.e., a stone which shows the direction. Later on this stone was used for navigation purposes. Work of William Gilbert showed that the earth itself behaves like a huge magnet and the directional property of lodestone was due to this reason. 

Magnet

A piece of substance which possesses the property of attracting small pieces of iron towards it is called a magnet. 

If the property of magnetism occurs naturally the magnet is known as a natural magnet. It is possible, artificially, to induce magnetism by rubbing the given piece of substance with magnets. The magnet, thus produced, is called an artificial magnet. 

Following terms are often, used in connection with a magnet:

1) Pole : It is a region of the magnet where it shows maximum magnetic characteristic. It is situated near the end of magnet. We shall denote it by m. Its unit, in S.I., is 'Am'. 

2) Magnetic axis : A line joining the two poles of the magnet is called magnetic axis. 

3) Magnetic meridian : A vertical plane passing through the axis of the magnet is called magnetic meridian. 

4) Magnetic length : Distance between the two magnetic poles is called magnetic length. It is denoted by '2l'. 

5) Magnetic moment : Product of pole strength of a magnet and its magnetic length is called magnetic moment. It is denoted by M (Rightward direction). 

              M = 2ml

It is a vector quantity. Its direction is from south pole to North Pole inside the magnet. In S.I., it is measured in 'Am²'.

Properties of a Magnet

A magnet possesses following properties :

1) Two poles of a magnet. A magnet has two poles. One is 'North seeking pole' or simply north pole (N) while the second is 'South seeking pole' or the south pole (S). These poles are situated a little distance inside the faces of the magnet (Fig. 1).

Fig. 1. Magnetic poles. 

Face to face length of the magnet is called geometric length while pole to pole length is called the magnetic length of the magnet. It is quite evident that the magnetic length is slightly less than geometric length. 

    Magnetic length, 2l = 7/8 × geometric length

2) Attracting property of a magnet. A magnet is capable of attracting small pieces of iron towards it. These pieces (iron filings) are attracted towards both the poles, north as well as south. This attraction is greatest at the poles and decreases as move towards the centre of magnet. It will be observed that the iron filings sticking to the magnet are largest in number near the poles (Fig. 2) and are smaller on the portion of magnet in between. 

Fig. 2. Magnetism strong at poles. 

3) Directional property of a magnet. When freely suspended a magnet always points in a particular direction. North Pole of the magnet points towards geographic north while south pole points towards the geographic south (Fig. 3).

Fig. 3. Suspended magnet points N-S. 

That is the reason these poles are called north seeking pole (or simply north pole) and south seeking pole (or simply south pole) respectively. The cause of this property is that earth behaves as a huge magnet which pulls the given magnet in a particular direction. This property led to the development of mariner's compass which is extensively used for navigation purposes. 

If the suspended magnet is deflected a bit from equilibrium position and released, it will execute torsional vibrations with N-S line as the mean position. Amplitude of vibrations will gradually decrease due to air resistance and the magnet will, ultimately, come to rest along N-S line. 

4) No existence of isolated magnetic poles. The magnetic poles exist only in pairs of opposite nature. It is not possible to obtain an isolated magnetic pole. If we break a magnet into two parts we get two independent complete magnets each with a pair of opposite pole. On further sub-division into four pieces, we shall obtain four complete magnets again with a pair of opposite poles (Fig. 4). This process continues till we reach the smallest particle i.e., an atom. The atom also behaves like a magnet. That is the reason we say that it is not possible to obtain an isolated pole. Sometimes, however, we need an isolated magnetic pole for theoretical consideration. In that case we can assume the pole of a very long magnet to be an isolated one, since the second pole will be situated at such a large distance away so as to be ineffective. 

Fig. 4. Breaking a magnet. 

5) Nature of force between two poles. Magnetic poles exert forces upon each other. The nature of force between similar poles is repulsive while that between opposite poles is attractive. 

This can be observed from following simple experiment.Bring the north Pole of magnet near the north Pole of a freely suspended magnet. The north Pole of suspended magnet also rotates away [Fig. 5(i)]. Similarly, it can be observed that south pole of suspended magnet also rotates away [Fig. 5(ii)] from the south pole, while the north pole of suspended magnet rotates towards [Fig. 5(iii)] the south pole of the magnet. This indicates that, "like poles repel each other while un like poles attract each other". This is called basic law of magnetistatics. 

Fig. 5. Magnetic attraction and repulsion. 

Coulomb's law of magnetistatics. It states that the magnitude of the force between two magnetic poles (supposed isolated) varies directly as the product of their pole strengths and inversely as the square of the distance between them. 

Consider two magnetic poles of similar nature (n) of strength m₁ and m₂ separated a distance r from each other (Fig. 6). The force (of repulsion) between them. 

                F ∝ m₁m₂ 

                F ∝ 1/r²         or        F ∝ m₁m₂/r²

or            F = k × m₁m₂/r²

where 'k' is the constant of proportionality. 

Fig. 6. Force between two magnetic poles. 

In S.I.,             k = μ₀/4π

∴                      F = μ₀/4π × m₁m₂/r²        ... (1) 

where μ₀ = 4π × 10-7  Wb A-1 m-1 

μ₀ is called the "absolute magnetic permeability" of free space. 

In C.G.S. system, k = 1

∴                            F = m₁m₂/r²               ... (2) 

Equation (1) and (2) represent mathematical forms of Coulomb's law of magnetostatics in S.I. and C.G.S. system respectively. 

6) Unit pole. Coulomb's law in magnetism, can lead to the definition of a unit pole. 

                     F = μ₀/4π × m₁m₂/r²

If m₁ = m₂, r = 1 m and F = 10-7  N and (we know) μ₀ = 4π × 10-7 Wb A-1 m-1    

Substituting these values, we get

                      10-7 = 4π × 10-7/4π  × m²/1

or                   m² = 1        or         m = ±1

So, a unit pole is that pole which when placed, in vaccum, at a distance of one metre from a similar pole repels it with a force 10-7 N. 

Repulsion is the sure test of magnetism

Supposing you are in possession of an iron bar and you want to know whether it is magnetised Or not. If it is magnetised it can show attraction towards a dissimilar pole of suspended magnet. If it is not magnetised it will again again show attraction with either of the magnetic poles. To be sure whether it is magnetised or not, we have to rely upon the repulsive characteristics of magnetic pole. 

     Bring the same end of the given bar near the two poles of the suspended magnet. It indicates attraction at one end repulsion at the other, it is magnetised. If it shows attraction at both ends it may be an unmagnetised iron bar. It will never show repulsion at both the ends. 

     Hence, "repulsion is the sure test of magnetism".

Important Notes

1. There is no existence of isolated magnetic poles. 
2. Magnetic poles are situated slightly inside the edges of magnet. Therefore, magnetic length is slightly less than geometric length. 
3. Strength of the two poles are always equal. 
4. Like poles are repel each other while unlike poles are attract each other. 
5. When freely suspended a magnet always points along N-S direction. 
6. Force of attraction/repulsion between two poles obeys inverse square law. 
7. Magnetic force between two poles is a central force. 

Atomic/Molecular Theory Of Magnetisation

Every atom of a magnetic substance is a magnetic dipole. In an unmagnetised piece, these magnetic dipoles form closed chains [Fig. 7(i)], thus neutralising each other's effects. The process of magnetisation consists in arranging these dipoles in regular manner, so that their magnetic moments are directed in one direction as shown in [Fig. 7(ii)]. 

Fig. 7. Effect of magnetic field on matter. 

As a result of this, one face of the specimen acquires a north polarity and the other one acquires a south polarity. The specimen is said to be magnetised and is called a magnet. The resultant magnetic moment 'M' of the magnet is given by

                  M = 2 ml

where     m = pole strength of the magnet

                 2l = distance between the poles

     Magnetic moment is a vector quantity. Its direction is from south to the north pole. 

Evidence Favouring Molecular Theory

Following experimentally observ

ed facts established the truth of molecular theory. 

     (i) Evidence of the two poles. During magnetisation the strength of the two poles always remains equal. This is due to the reason that alignment of one dipole results in liberation of one free north and one south pole. Greater the number of dipoles arranging themselves in regular manner, greater is the strength of magnet. 

     (ii) Magnetic saturations. When the manetising field is so large that all the dipoles have aligned themselves along the field, further efforts for magnetisation cannot increase the strength of magnet beyond that value. The magnet is said to have achieved saturation stage. 

     (iii) Magnetic induction. When a piece of soft iron is placed in the neighbourhood of a magnet, a pole of opposite nature is induced on its nearer end while a pole of similar nature is induced on further end. The north pole of the magnet attracts the south poles of the dipoles of soft iron towards it, thus making the nearer end south end farther end north. 

     (iv) Demagnetisation. It has been observed that due to rough handling or heating, a magnet loses its magnetism. This is due to the reason that heating results in an increase in kinetic energy of atoms. As they vibrate more vigorously, they form closed chains again, thus losing the magnetism.