# Magnetism

A bar magnet consists of two equal and opposite magnetic poles, separated by a distance; hence a magnet is also called a magnetic dipole.

If ‘m’ is the pole strength and 2l is the separation between the poles, then magnetic moment of the bar magnet, $\overrightarrow{M} = m \overrightarrow{2l}$ . The magnet moment is a vector. Its direction is from –m to +m.

Bar Magnet

#### Non-existence of free magnetic poles

If a magnet is cut in two halves from the middle then –m and +m poles cannot be separated; but each half becomes magnet with its magnetic moment M/2.

Non-existence of free magnetic poles

If a magnet of magnetic moment M is divided into n equal parts, the n each part is a magnet with magnetic moment $\dfrac{M}{n}$ . Accordingly smallest part of a magnet is complete magnetic dipole. According to molecular theory of magnetism, each molecule of a magnetic material is a tiny bar magnet or a magnetic dipole.

#### Equivalence of a magnetic dipole with a current loop

A small current carrying conductor of area A and carrying current ‘I’ is equivalent to a magnet of magnetic moment:

M=N I A

Where N=number of turns in the loop. Obviously the unit of magnetic moment is $\text{Ampere-meter} ^2$ . The direction of magnetic moment is along the normal to the plane of the loop.

Remark: The unit of pole strength is Ampere X meter (A – m). The pole strength depends on the number of molecules in the cross sectional area.

#### Comparison of a Bar magnet with a solenoid

A current carrying solenoid is equivalent to a bar magnet. When a bar magnet is suspended freely from its mid-point, it say along north-south direction. Similarly when a current carrying solenoid is suspended freely, it also stays along north-south direction.

North and South-poles of a solenoid: When a current is passed in a solenoid it acts as a bar magnet. If the current in the coil of nearer face of solenoid is anticlockwise, the end  is ‘North Pole”; but if it is clockwise, the nearer face is ‘South-Pole”.

North and south poles of solenoid

#### Magnetic lines of Force

The magnetic lines of force are the imaginary lines which continuously represent the direction of magnetic field.

Hypothetically a magnetic line of force is the line/curve in which an isolated north pole initially lying on it at rest would travel. But isolated magnetic poles do not exist; therefore magnetic lines of force are described by a small compass needle. A compass needle sets itself with its magnetic axis in the direction of magnetic field at that point.

Magnetic field in term of magnetic lines of force: The number of magnetic lines of force passing per unit area normally around a given point is a measure of magnitude of the magnetic field. Accordingly if the lines of force are crowded, the magnetic field is stronger and if they are farther, the magnetic field is weaker.

The tangent drawn on a line of force at any point gives the direction of magnetic field at that point.

##### A few properties of magnetic lines of force:

(i) The magnetic lines of force are always closed curves. For a bar magnet they start from north pole and following a curved path enter the south pole and inside the magnet they continue to run from south pole to north pole.

Magnetic lines

(ii) Magnetic lines of force never intersect; because if they would intersect, there will be two tangents at the point of intersection which means two directions of magnetic field at the same point, which is impossible.

(iii) If magnetic field is uniform, the magnetic lines of force are equidistant.

#### Torque on a Bar Magnet in a magnetic field

If a bar magnet is placed in a uniform magnetic field B, its poles +m and –m experience force mB and mB along and opposite to the direction of magnetic field B; so net force on the bar magnet is zero.

Torque on a bar magnet

i.e.

F = mB – mB =0

But the force being equal and opposite and having separation between their lines of action form a couple. The moment of forces or couple is given by:

$\tau = Force \times \text{perpendicular distance}$ $= ( mB ) ( bn ) = m B . 2 l sin \theta$

i.e.

$\tau = m 2 l$ is magnetic moment of magnet.

In vector form $\overrightarrow{ \tau} = \overrightarrow{M} \times \overrightarrow{B} newton - meter$

If $\theta = 9060$ , the torque is maximum given by :

$\tau _{max} = M B \rightarrow M = \dfrac{\tau _{max}}{B}$

##### Potential energy of a magnetic dipole in a uniform magnetic field

: If a magnetic dipole if moment M is placed in a uniform magnetic field B making an angle $\theta$ with the magnetic field, then,

$Potential \, \, energy \, \, U = - M B cos \theta$

In vector form U = $- \overrightarrow{M} \times \overrightarrow{B}$

Work done in rotating the dipole from equilibrium position through an angle $\theta$ :

$W = M B ( 1 - cos \theta )$

#### Magnetic field due to a bar magnet

(i) at axial position (p):

$B = \dfrac{\mu _0}{4 \pi} \dfrac{2M}{r^3}$

(ii) at equatorial position (Q)

$\dfrac{\mu _0}{4 \pi} \dfrac{M}{r^3}$

#### Force between two short bat magnets (at distance r apart):

(i) When magnets are co-axial:

$F = \dfrac{\mu _0}{4 \pi} \dfrac{6 M_1 M_2}{r^4}$

Force between two short magnet

(ii) When magnets are perpendicular:

$F = \dfrac{\mu _0}{4 \pi} \dfrac{3 M_1 M_2}{r^4}$

#### Earth’s Magnetic Field

The earth behaves as a magnet. When a magnet is suspended freely, it says along north-south direction. The north and south poles of magnet stay along north and south poles of earth respectively. The magnetic poles are at some distance from geographical poles.

The latest theories of earth’s magnetism are:

(i) The earth rotates about its axis and has the surrounding ionized region due to interaction of cosmic rays. Due to rotation of earth the surrounding ionized region gives to strong a electric current which causes magnetism.

(ii)There exist molten iron and nickel within the core of earth. When earth rotates it behaves as a dynamo and causes magnetism.

Accordingly to both theories given above, the earth’s magnetism with the destroyed if it stops rotating.

#### Components of Earth’s magnetic Field:

(a) Angle of declination $\varphi$ : The magnetic field of earth varies in magnitude and direction.

The angle of dip is the angle made by resultant earth’s magnetic field with the horizontal.

The angle of dip is measured by dip circle.

At poles of dip is measured by dip circle.

At poles $\theta = 90^0$ and at equator $\theta = 0^0$ .

Angle of declination

(c) Horizontal components of Earth’s Magnetic field: The earth’s field strength be Be may be resolved in two components

(i) Horizontal components (H)

(ii) Vertical components V

From fig $H = B_e cos \theta \, V = B_e sin \theta$ these equation give:

$B_e = \sqrt{H^2 + V^2}$

$tan \theta = \dfrac{V}{H}$

#### Element of magnetic maps:

(i) Isogonic lines: There are the lines joining the place of same declination.

(ii) Agonic lines: These are the lines joining the places of zero declination.

(iii) Isoclinic lines: These are the lines joining the points of equal dip.

(iv) Actinic lines or magnetic Equator: These are the lines joining the points of zero dip.

(v)Isodynamic lines: these are the lines joining the places of equal ‘H’

#### Tangent law

If two horizontal magnetic fields $B_1$ and $B_2$ act perpendicular to each other, then their resultant makes an angle $\theta$ with $B_1$ such that:

$tan \theta = \dfrac{B_2}{B_1}$

Tangent Law

This is called tangent law. If a compass needle is placed at any point in such a field, it stays pointing along the resultant.

#### Tangent Galvanometer

Tangent galvanometer is a device to measure the current based on tangent law. In this galvanometer the two perpendicular magnetic fields are:

(i) Horizontal components of earth’s magnetic field H pointing from south to north.

(ii) A horizontal magnetic field B, pointing in east-west direction, produced by passing a current in a circular coil mounted in vertical plane in north south direction. The compass needle is placed at the center of the coil and stays along the resultant magnetic field $B_r$ .

According to tangent law,

$tan \theta = \dfrac{B}{H} \rightarrow B = H tan \theta$

Tangent law

If ‘r’ is the radius of coil, n – number of turns in the coil and I the current flowing in coil, then magnetic field at center of coil.

$B = \dfrac{\mu _0 Ni}{2r}$

Therefore,

$\dfrac{ \mu _0 N I}{2r} = H tan \theta$

Or,

$I = \dfrac{2r H}{\mu _0 N} tan \theta$

For given coil r, N are constant and at a particular place horizontal components of earth’s magnetic field is constant.

$I = K tan \theta$

Where $K = \dfrac{2r H}{\mu _0 N} = constant$ , called the reduction factor of tangent galvanometer.

Obiviously $I \propto tan \theta$

i.e. the current flowing is proportional to the tangent of the deflection of the compass needle.

The quality $\dfrac{tan \theta}{i} = \dfrac{\mu _0 N}{2r H}$ is a measure of sensitivity of galvanometer.

For greater sensitivity:

(i) The radius r should be small

(ii) Number of turns N should be large

(iii) Earth’s magnetic field should be low

(iv) With small variation of I, $tan \theta$ should vary rapidly.

As $tan \theta$ varies rapidly near 45 degree; therefore sensitivity of galvanometer is maximum when deflection is near 45 degree.

#### Vibration magnetometer

The vibration magnetometer is used to compare magnetic moments of two magnets. Suppose a bar magnet of magnetic moment M oscillates in a horizontal magnetic field (H) of earth. Let $\theta$ be the instantaneous deflection. The torque acting on the magnet $\varphi = MH sin \theta$ . This torque tends to bring the magnet in mean position. Therefore it is restoring torque i.e. $\varphi = - M H sin \theta$

Vibration magnetometer

If I is the moment of inertia of rotating system and $\alpha$ the acceleration, ie

$I \alpha = - M H sin \theta \rightarrow = - \dfrac{MH}{I} sin \theta$

If deflection of magnet is small, $sin \theta = \theta$

Therefore,

$\alpha = - \dfrac{MH}{I} \theta$

I.e.

$\overrightarrow{\alpha} \propto - \overrightarrow{\theta}$

i.e.

motion of magnet is angular SHM

Standard equation of angular SHM is $\alpha = - \omega ^2 \theta$

Time period of motion,

$T = \dfrac{2 \pi}{\omega} = 2 \pi \sqrt{\dfrac{I}{MH}}$

Case (I) : If magnet oscillates in horizontal plane in magnetic meridian (N – S direction), it oscillates under horizontal magnetic field.

i.e

$T = 2 \pi \sqrt{\dfrac{I}{MH}}$

Case (II): If magnet oscillates in vertical plane perpendicular to magnetic meridian; it oscillates under vertical magnetic field:

I.e.

$T = 2 \pi \sqrt{\dfrac{I}{MV}}$

Case (III): If magnet oscillates in a vertical plane parallel to the magnetic meridian, if oscillates under total magnetic field of earth.

i.e

$T = 2 \pi \sqrt{\dfrac{I}{MB_e}}$

Comparison of magnetic moments:  Let two magnets have magnetic moments $M_1$ and $M_2$ . if $T_1$ is the time period with two combined magnets with their poles concluding and $T_2$ the time period when their unlike poles concluding; then

$\dfrac{M_1}{M_2} = \dfrac{T_1 ^2 + T_2 ^2}{T_2 ^2 - T_1 ^2}$

Comparison of magnetic fields: If $B_1$ and $B_2$ are the magnetic fields produced by two magnets when put with the magnets of magnetometer successively and T is the time period of magnetometer magnet under the earth’s magnetic field alone, then:

#### Some definitions

When a substance is placed in magnetic field ‘H’, it gets magnetized and attains magnetic moment ‘M’.

Intensity of magnetization: The magnetic moment per unit volume of the substance is called the intensity of magnetization.

I.e.

$I = \dfrac{M}{V} amp/meter$

In the case of a bar magnet M= m.2l

V = A. 2l

Therefore,

$I = \dfrac{m 2l}{A 2l} = \dfrac{m}{A}$

I.e. intensity of magnetization is also equal to pole strength per unit area.

Magnetic Permeability: The magnetic permeability is defined as the ratio of magnetic induction in material to the magnetizing field.

I.e.

$\mu = \dfrac{B}{H} \rightarrow = B = \mu H$

Magnetic Susceptibility: for most of materials the magnetization induced is linearly related to magnetizing field.

The ratio of intensity of magnetizing field is called the magnetic susceptibility.

i.e.

$X_m = \dfrac{I}{H}$

Relation between B, H and I: The magnetic induction within the substance is the vector sum of the magnetizing field applied and magnetization induced in proper units.

i.e.

$B = \mu _0 ( H + I )$

$B = \mu H$

$\mu H = \mu _0 ( H + I )$

$or \, \dfrac{\mu}{\mu _0} = 1 + \dfrac{I}{H}$

As $\dfrac{\mu}{\mu _0}$ = relative permeability of material = $\mu _r$ and $\dfrac{I}{H} = X_m$

Therefore,

$\mu _r = 1 + X_m$

#### Dia, para and ferromagnetic substance

According to behavior of the substance in the magnetic field, they are classified into three categories:

(1) Diamagnetic substance: The substances which when placed in a strong magnetic field acquire feeble magnetization opposite to the direction of the magnetic field are called diamagnetic substance. The examples are copper, gold, antimony, bismuth, alchol, water, quartz, hydrogen Nacl etc

Characterstics:

(i) They are repelled by a strong magnet

(ii) The magnetic properties such as magnetic moment, susceptibility, intensity of magnetization are negative and small.

(iii) The relative permeability $\mu _r$ is less than unity.

(iv) In a non-uniform magnetic field they experience attraction towards weaker parts of the magnetic field.

Diamagnetic substance

(v) A rod of a diamagnetic substance when suspended between poles pieces of a magnet stays with its axis perpendicular to the magnetic field produces by poles.

Diamagnetic substance beetwen two poles

(vi) When a strong magnetic field is applied across one limb of U-tube with a diamagnetic liquid; the liquid in that limb is dispersed.

(viii) The magnetic susceptibility of diamagnetic substances is independent of the magnetizing field and the temperature.

Origin: The diamagnetism is usually found in those substances whose atoms / molecules have even number of electrons which form pairs of opposite spin. So magnetic moment of electron is neutralized by the other; so in the absence of any magnetic field, the magnetic moment of diamagnetic substances is zero. As electrons of all substances have tendency to form pairs of opposite spins. The diamagnetism is the universal property of all substances.

When a diamagnetic substance is placed in an external strong magnetic field, then one electron of pair is accelerated and the other is retarded; so that magnetism is included in opposite direction.

#### Paramagnetic substances

The substances which when placed in a strong magnetic field acquire feeble magnetization in the direction of the applied magnetic field, are called paramagnetic substances. The examples are platinum, aluminum, chromium, sodium, $CuSo_4 \, O_2$ etc.

Characteristics:

(i) They are attracted by a strong magnet

(ii) The magnetic properties such as magnetic moment, magnetic susceptibility, intensity of magnetism are positive but small.

(iii) The relative permeability is slightly greater than unity.

(iv) In a non-uniform magnetic field, the paramagnetic substance is attracted towards stronger part of the field.

(v) When a rod of paramagnetic substance is suspended freely between the strong poles of a magnet, then its axis becomes parallel to the magnetic field.

(vi) When a strong magnetic field is applied across one limb of U-tube filled with a paramagnetic liquid, the level of liquid in that limb rises.

(vii)The susceptibility and permeability do not change with the variation of magnetizing field.

(viii) The susceptibility of a paramagnetic substance is inversely proportional to the absolute temperature.

I.e.

$X_m \propto \dfrac{1}{T}$

This is called curie law.

Origin: Para magnetism result due to excess electrons spinning in the same direction.

#### Ferromagnetic substance

The substances which when placed in an external magnetic field acquire strong magnetization in the direction of applied magnetic field, are called ferromagnetic substance. The examples are cobalt, nickel, iron, gadolinium and their alloys.

Characteristics:

(i) They are even attracted by a weak magnet

(ii) The susceptibility in very large and positive

(iii) The relative permeability is very high

(iv) The intensity of magnetization is proportional to the magnetizing field H for smaller values, varies rapidly for moderate values and attains a constant value for larger values of H.

Origin: Ferromagnetism results due to strong forces of interaction arising due to exchange coupling among neighboring atoms and is usually explained by domain theory.

#### Hysteresis

The graph representing the variation of intensity of magnetization I of a ferromagnetic material versus magnetizing field ‘H’ is shown in the figure. The graph is called hysteresis curve because the intensity of magnetization lags behind the magnetizing field throughout.

The graphs shows:

Hysteresis

(i) When magnetizing field is increased. The intensity of magnetization increases and becomes maximum. The maximum value is called the saturation value.

(ii) When magnetizing field is decreased, the intensity of magnetizing decrease and when magnetizing field H = 0, the intensity of magnetization is not zero, but it is equal to Ob; this is called residual magnetism or retentively.

(iii) When magnetizing field is reversed in direction, the intensity of magnetizing decreased and at a particular value of reverse magnetic field, it becomes zero. The reverse field (Oc) is a measure if tolerance to retain the magnetizing with it and is called coercivity.

(iv) When reverse field is further increased, the substance attains saturation in reverse direction. When magnetic field is again changed in direction, we get symmetrical lower curve. The complete curve abcdea is called the hysteresis cycle or hysteresis loop.

The area of hysteresis loop gives a measure of energy loss in magnetizing and demagnetizing the ferromagnetic material.

#### Characteristics of soft iron and steel:

1. Soft iron: For soft iron the susceptibility, permeability and retentively are greater while coercitivity is low. So hysteresis cycle for soft iron is long and narrow. Hysteresis loss per cycle is small and hence soft iron is used for making electromagnets, cores of transformers, telephone diaphragms and armatures of generators, motors.

2. Steel: For steel the susceptibility and permeability are relatively lower than soft iron. The retentively of steel is also relatively smaller than soft iron, but coercitivity of steel is very high.

The hysteresis loop foe steel is long and wide therefore energy loss per cycle of magnetization and demagnetization is high.

#### Methods to destroy magnetism:

The magnetization of a magnet may be destroyed by:

(i) Heating it above critical temperature.

(ii)Passing alternating current

(iii) Applying reverse magnetic field of strength more than coercivity.

(iv) By rough handling i.e. dropping and hammering.

#### Neutral point

Neutral point is a point where net magnetic field is zero. At neutral point the horizontal components of earth’s magnetic field is balanced by the magnetic field produced by the external magnet.

When the north pole of a bar magnet points along the north direction, the neutral point is obtained at equatorial position of magnet, so at neutral point:

$H = \dfrac{\mu _0}{4 \pi} \dfrac{Md}{ ( d^2 + l^2 ) ^{ \dfrac{3}{2}}}$

When the North pole of the bar magnet points along the south-direction, the neutral point is obtained at the axis of magnet; so at neutral point:

$H = \dfrac{\mu _0}{4 \pi} - \dfrac{2Md}{ ( d^2 - l^2 ) ^2}$

If magnet is short l << r; then:

$H = \dfrac{\mu _0}{4 \pi} \dfrac{2M}{d^3}$

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