CBSE NCERT NOTES CLASS 12 PHYSICS CHAPTER 5

MAGNETRISM AND MATTER

Magnetism

The property of any object by virtue of which it can attract a piece of iron or steel is called magnetism.

1. The poles of a magnet are the two points near but within the ends of the magnet, at which the entire magnetism can be assumed to be concentrated.

2. The poles always occur in pairs and they are of equal strength. North and south poles of a magnet cannot be separated.

3. Like poles repel and unlike poles attract.

4. A freely suspended magnet always aligns itself into north-south direction.

Properties of magnetic field lines

1. The magnetic field lines of a magnet (or a solenoid) form continuous closed loops. This is unlike the electric dipole where these field lines begin from a positive charge and end on the negative charge or escape to infinity.

2. They emerge from north pole and merge at south pole outside the magnet. The direction is from south to north pole inside the magnet.

3. The tangent to the field line at a given point represents the direction of the net magnetic field B at that point.

4. The larger the number of field lines crossing per unit area, the stronger is the magnitude of the magnetic field B. The field is stronger near the magnet and weaker as we move away.

5. The magnetic field lines do not intersect, for if they did, the direction of the magnetic field would not be unique at the point of intersection.

Magnetic dipole

Magnetic dipole is an arrangement of two unlike magnetic poles of equal pole strength separated by a very small distance, e.g., a small bar magnet, a magnetic needle, a current carrying loop etc.

Magnetic dipole moment

The product of the distance (2l) between the two poles and the pole strength of either pole is called magnetic dipole moment.

Magnetic dipole moment m = qm (2l).

Its SI unit is ‘Joule/Tesla’ or A m2.

Its direction is from south pole to north pole.

Coulomb’s law for magnetism

The force of interaction acting between two magnetic poles is directly proportional to the product of their pole strengths and inversely proportional to the square of the distance between them.

where

qm1 and qm2 = pole strengths,

r = distance between poles and

μo = permeability of free space.

Combination of magnets

• When a magnet having pole strength qm and magnetic moment m is cut into two equal parts
1. Parallel to its length

2. Perpendicular to its length

• When a magnet of length l, pole strength qm and of magnetic moment m is turned into a semicircular arc then it new magnetic moment

$\mathrm{m}\mathrm{\text{'}}={\mathrm{q}}_{\mathrm{m}}×2\mathrm{r}=\frac{2\mathrm{m}}{\mathrm{\pi }}$

BAR MAGNET AS AN EQUIVALENT SOLENOID

The magnetic dipole moment m associated with a current loop is m = NIA, where N is the number of turns in the loop, I the current and A the area vector.

Let us assume a solenoid of length 2l, radius a, consisting of n turns per unit length, carries a current I.

Consider a circular element of thickness dx of the solenoid at a distance x from its centre.

It this element consists of ndx turns.

The magnetic field due to this element on the axis of a solenoid at a distance r from its centre will be,

The magnitude of the total field is obtained by summing over all the elements from x = -l to x = +l

For r >> a, and r >> l

So,

Putting m = NIA = n 2l I πa2

We get,

The dipole in a uniform magnetic field

Therestoring torque

In equilibrium

where $\mathcal{I}\mathbf{}$ = moment of inertia

For small deflections θ = sin θ,

Therefore,

This represents a simple harmonic motion. The square of the angular frequency is

Therefore the time period will be,

The magnetic potential energy Um is given by the work done in rotating the magnet by angle θ,

We can see that

• Potential energy is minimum (= -mB) at θ = 0 (most stable position)

• Potential energy is maximum (= +mB) at θ = 180 (most unstable position).

Comparision between electric dipole and magnetic dipole

 Characteristic Electrostatics Magnetism $\frac{1}{{\mathrm{\epsilon }}_{\mathrm{o}}}$ ${\mathrm{\mu }}_{\mathrm{o}}$ Dipole moment $\stackrel{\to }{\mathrm{m}}$ Equatorial Field for a short dipole $–\frac{\stackrel{\to }{\mathrm{p}}}{4\mathrm{\pi }{\mathrm{\epsilon }}_{\mathrm{o}}{\mathrm{r}}^{3}}$ Axial Field for a short dipole $\frac{2\stackrel{\to }{\mathrm{p}}}{4\mathrm{\pi }{\mathrm{\epsilon }}_{\mathrm{o}}{\mathrm{r}}^{3}}$ External Field: torque External Field: Energy $–\stackrel{\to }{\mathrm{p}}.\stackrel{\to }{\mathrm{E}}$ $–\stackrel{\to }{\mathrm{m}}.\stackrel{\to }{\mathrm{B}}$

Gauss’s law for magnetism

Net magnetic flux through a closed surface is zero.

Consider a small vector area element Δ$\stackrel{\to }{\mathrm{S}}$ of a closed surface S.

The magnetic flux through Δ$\stackrel{\to }{\mathrm{S}}$ is defined as ΔФB = $\stackrel{\to }{\mathrm{B}}.\mathrm{\Delta }\stackrel{\to }{\mathrm{S}}$, where $\stackrel{\to }{\mathrm{B}}$ is the field at $\mathrm{\Delta }\stackrel{\to }{\mathrm{S}}$.

Lt us divide S into many small area elements and calculate the individual flux through each. Then, the net flux ФB is,

Surface integral of magnetic field over any closed surface is always 0.

$\oint \stackrel{\to }{\mathrm{B}}.\mathrm{d}\stackrel{\to }{\mathrm{S}}=0$

EARTH’S MAGNETISM

Earth is a huge magnet. The strength of the earth’s magnetic field varies from place to place on the earth’s surface; its value being of the order of 10–5 T.

Cause of earth’s magnetism

The magnetic field is arises due to electrical currents produced by convective motion of metallic fluids (consisting mostly of molten iron and nickel) in the outer core of the earth. This is known as the dynamo effect.

The earth’s magnetic field lines enter through the magnetic north and leave through the magnetic south.

The axis of the dipole of the earth does not coincide with the axis of rotation of the earth but is presently titled by approximately 11.3o with respect to the later.

The magnetic poles are located where the magnetic field lines due to the dipole enter or leave the earth.

The north magnetic pole is at latitude of 79.74o N and a longitude of 71.8o W, a place somewhere in north Canada. The south magnetic pole is at 79.74o S, 108.22o E in the Antarctica.

The pole near the geographic north pole of the earth is called the north magnetic pole and the pole near the geographic south pole is called south magnetic pole.

The true magnetic poles are opposite of the convention.

Magnetic Meridian is a vertical plane passing through the magnetic axis of the earth. This is perpendicular to the magnetic equatorial plane.

Geographic Meridian is a vertical plane passing through the geographic axis. This is perpendicular to the geographic equatorial plane.

There are three components of earth’s magnetism at any point,

1. Magnetic Declination (θ or D) - Magnetic declination is the smaller angle subtended between the magnetic north (the direction the north end of a compass needle points) and true north. The declination is positive when the magnetic north is east of true north.

2. Magnetic Inclination or Magnetic Dip (δ or I) - Magnetic inclination is the angle made by a compass needle when the compass is held in a vertical orientation. Positive values of inclination indicate that the field is pointing downward, into the Earth, at the point of measurement.

3. Horizontal Component of Earth’s Magnetic Field (H) - If B is the intensity of earth’s magnetic field then horizontal component of earth’s magnetic field

H = B cos δ

Vertical component of earth’s magnetic field

V = B sin δ

Hence,

And

• Angle of dip is zero at magnetic equator and 90° at poles.

Magnetisation and magnetic intensity

A circulating electron in an atom has a magnetic moment. In a bulk material, these moments add up vectorially and they can give a net magnetic moment which is non- zero.

Magnetisation M of a sample is equal to its net magnetic moment per unit volume:

The magnetic field in the interior of the solenoid is given by

Bo = μo n I

⇒ Bo = μo H

Where H is the magnetic intensity inside the solenoid

If the interior of the solenoid is filled with a material with non-zero magnetisation, the field inside the solenoid is found to be

B = Bo + Bm

where Bm is the field contributed by the material core.

It has been found that Bm is proportional to the magnetisation M of the material and is expressed as

Bm = μo M

Also magnetic intensity

Where H is the magnetic intensity due to solenoid and M is due to the nature of core. M is dependent on H as per following relation,

where 𝜒 is a dimensionless quantity called the magnetic susceptibility of the material.

Now,

Where μr = (1+ 𝜒) is a dimensionless quantity called the relative magnetic permeability of the substance.

Magnetic permeability

CLASSIFICATION OF MATTER BASED ON MAGNETIC PROPERTIES

Diamagnetic substances

Diamagnetic substances are those which have tendency to move from stronger to the weaker part of the external magnetic field. They move from region of strong magnetic field to weak magnetic field, i.e., they get repelled by a magnetic field.

Examples - Bismuth, copper, lead, silicon, nitrogen (at STP), water and sodium chloride.

Explanation of diamagnetism

Electrons in an atom orbiting around nucleus possess orbital angular momentum and are equivalent to current-carrying loop and thus possess orbital magnetic moment.

Diamagnetic substances are the ones in which resultant magnetic moment in an atom is zero.

When magnetic field is applied, those electrons having orbital magnetic moment in the same direction slow down and those in the opposite direction speed up. This happens due to induced current in accordance with Lenz law, Thus, the substance develops a net magnetic moment in direction opposite to that of the applied field and hence the repulsion.

Superconductors and diamagnetism

Superconductors are metals, cooled to very low temperatures which exhibits both perfect conductivity and perfect diamagnetism (𝜒 = -1, μr = 0). A superconductor repels a magnet. Here the field lines are completely expelled out of the material!

Diamagnetism in superconductors is called the Meissner effect.

Superconducting magnets are used in variety of situations, for example, for running magnetically levitated superfast trains.

Paramagnetic substances

Paramagnetic substances are those which get weakly magnetised when placed in an external magnetic field. They have tendency to move from a region of weak magnetic field to strong magnetic field, i.e., they get weakly attracted to a magnet.

The individual atoms (or ions or molecules) of a paramagnetic material possess a permanent magnetic dipole moment of their own. On account of the ceaseless random thermal motion of the atoms, no net magnetization is seen. In the presence of an external field Bo, which is strong enough, and at low temperatures, the individual atomic dipole moment can be made to align and point in the same direction as Bo

The field lines get concentrated inside the material, and the field inside is enhanced.

Examples - aluminium, sodium, calcium, oxygen (at STP) and copper chloride.

Curie’s law of parmagnetism

Magnetisation of a paramagnetic material is proportional to the external field intensity Bo inversely proportional to the absolute temperature T,

Bo = μo H

And

M = χH

Here C is called Curries constant.

Ferromagnetic substances

In a ferromagnetic substance, there are several tiny regions called domains. Each domain contains approximately 1011 atoms.

Each domain is a strong magnet as all atoms or molecules in a domain have same direction of magnetic moment.

In the absence of any external magnetic field, the domains are randomly oriented.

When we apply an external magnetic field Bo, the domains orient themselves in the direction of Bo and simultaneously the domains oriented in the direction of Bo grow in size, resulting in the net magnetic dipole moment.

In a ferromagnetic material the field lines are highly concentrated. In non-uniform magnetic field, the sample tends to move towards the region of high magnetic field.

Hard ferro magnets

In some ferromagnetic materials the magnetisation persists after removal of the external field. Such materials are called hard magnetic materials or hard ferromagnets. Alnico, an alloy of iron, aluminium, nickel, cobalt and copper, is one such material. The naturally occurring lodestone is another. Such materials form permanent magnets.

Soft ferro magnets

Ferromagnetic materials, in which the magnetisation disappears on removal of the external field, are called soft ferro-magnets. Example - soft iron, cobalt, nickel, gadolinium, etc. The relative magnetic permeability is >1000!

Temperature dependence of ferromagnetic property

The ferromagnetic property depends on temperature. At high enough temperature, a ferromagnet becomes a paramagnet. The domain structure disintegrates with temperature. The temperature of transition from ferromagnetism to paramagnetism is called the Curie temperature TC.

$\mathrm{\chi }=\frac{\mathrm{C}}{\mathrm{T}-{\mathrm{T}}_{\mathrm{c}}}$

• In terms of the susceptibility χ , a material is diamagnetic if χ is negative, para- if χ is positive and small, and ferro- if χ is large and positive.

Here ε is a small positive number introduced to quantify paramagnetic materials.

 Diamagnetic Paramagnetic Ferromagnetic

Hysteresis

The lagging of intensity of magnetisation (I) or induced magnetic field (B) behind magnetising field (H), when a specimen of a magnetic substance is taken through a complete cycle of magnetisation is called hysteresis.

The relation between B and H in ferromagnetic materials is not linear and it depends on the magnetic history of the sample.

1. Let us place the unmagnetised sample in a solenoid and increase the current through the solenoid. The magnetic field B in the material rises and saturates as depicted in the curve Oa. This behavior represents the alignment and merger of domains until no further enhancement is possible. The saturated magnetic field BS ~ 1.5 T.

2. Next, we decrease H and reduce it to zero. At H = 0, B ≠ 0. This is represented by the curve ab. The value of B at H = 0 is called retentivity or remanence, represented by BR ~ 1.2 Tesla. The domains are not completely randomised even though the external driving field has been removed.

3. Next, the current in the solenoid is reversed and slowly increased. Certain domains are flipped until the net field inside stands nullified. This is represented by the curve bc. The value of H at c is called coercivity, represented by HC.

4. As the reversed current is increased in magnitude, we once again obtain saturation. The curve cd depicts this. The saturated magnetic field ~ - 1.5 T

5. Next, the current is reduced (curve de) and reversed (curve ea). The cycle repeats itself. Note that the curve Oa does not retrace itself as H is reduced. For a given value of H, B is not unique but depends on previous history of the sample.

Retentivity or residual magnetism

The value of the intensity of magnetisation of a material, when the magnetising field is reduced to zero is called retentivity or residual magnetism of the material.

Coercivity

The value of the reverse magnetising field that should be applied to a given sample in order to reduce its intensity of magnetisation or magnetic induction to zero is called coercivity.

• Permanent magnets

Commonly steel is used to make a permanent magnet because steel has high residual magnetism and high coercivity.

Permanent magnets are made by the materials such as steel, for which

• residual magnetism is high

• coercivity is high.

• Electromagnets

Core of electromagnets are made of soft iron because area of hysteresis loop for soft iron is small. Therefore, energy loss is small for a cycle of magnetisation and demagnetisation.

Electromagnets are made by the materials such as soft iron for which

• permeability is high,

• residual magnetism is low,

• coercivity is low and

• hysteresis loss is low.

Tangent law for magnetism

When a bar magnet is freely suspended under the combined effect of two uniform magnetic fields of intensities B and H acting at 90° to each other, then it bar magnet comes to rest making an angle θ with the direction of H, then

B = H tan θ