Magnetism And Matter:Revision Notes

Ruhul Amin Alig

A quick revision of all the important concepts

1. The Magnetic Field Lines

Magnetic field lines form continuous closed curves.Magnetic field lines have no source or sink.
The tangent drawn at any point on field line represents direction of the field at that point. Field lines never intersect to each other.
At any place crowded lines represents stronger field while distant lines represent weaker field.
In any region, if field lines are equidistant and straight the field is uniform otherwise not.

2. Bar Magnet

Magnetic dipole consists of two unlike poles of equal strength separated by a distance. Bar magnet is an example of a magnetic dipole.

Properties of a bar magnet:

A freely suspended bar magnet will always point towards the North- South Direction.
Like magnetic poles repel and unlike ones attract.
Magnetic poles cannot be isolated i.e., they always exists in pair
When a magnet is broken, each piece behave like a magnet with a pair of North and South Pole
The Pole strength of each pole of a magnet is equal.

2.a Dipole Moment of a Bar Magnet

The magnetic dipole moment of a bar magnet is defined as the product of its pole strength and magneticlength.

Magnetic Moment

M = q_{m} d

where

d

is a vector pointing from South to North pole of the bar magnet, the magnitude of

d

is equal to the magnetic length of the bar magnet.

∣ d ∣ = 2 l

2.b Magnetic Length and Geometric Length

The length of the bar magnet is called geometrical length and the length between two magnetic poles in a bar magnet is called magnetic length. Magnetic length is always slightly smaller than geometrical length.

\frac{M a g n e t i c L e n g t h}{G e o m e t r i c L e n g t h} = \frac{5}{6}

2.c Magnetic Field due to a bar magnet

Magnetic field due to a bar magnet at an axial point

B_{a x i a l} = \frac{μ _{o}}{4 π} \frac{M r}{( r ^{2} - l ^{2} ) ^{2}}

Magnetic field due to a bar magnet at an equatorial point -

For a small bar magnet, l<

B_{a x i a l} = \frac{μ _{o}}{4 π} \frac{M}{( r ) ^{3}}

B_{e q u i} = - \frac{μ _{o}}{4 π} \frac{M}{( r ^{2} + l ^{2} ) ^{3 / 2}}

for small bar magnet, l<

B_{e q u i} = - \frac{μ _{o}}{4 π} \frac{M}{( r ) ^{3}}

2.d Torque on a bar magnet in an uniform magnetic field

When a bar magnet is kept within a magnetic field, both the poles experiences equal and opposite forces (

q_{m} B

).These two equal and opposite forces constitute a couple (about midpoint of bar magnet) tend to align the magnet in the direction of the magnetic field

B

Torque experienced by the bar magnet is given by

τ = M \times B = M B s i n θ

, where

M

is the magnetic dipole moment.

2.e Potential energy of a magnetic dipole in a uniform magnetic field

U = \int_{θ_{1}}^{θ_{2}} τ d θ = \int_{θ_{1}}^{θ_{2}} M B s i n θ d θ = - M B [c o s θ]_{θ_{1}}^{θ_{2}} = - M . B {c o s θ_{2} - c o s θ_{1}}

Taking final angle to be 90 degrees we can say,

U = - M . B = M B c o s θ

potential energy of the bar magnet is minimum when it is aligned along the external magnetic field and maximum when the bar magnet is aligned anti-parallel to external magnetic field.

2.f Oscillation of a freely suspended magnet -

T = 2 π \frac{I}{M B}

where I: moment of inertia

3. Magnetism and Gauss's law :

The net magnetic flux through a closed surface is zero.

\oint B . d s = 0

This leads to conclusion that magnetic poles cannot exist separately

4. Analogy between electric and magnetic dipoles :

5. Earth's magnetic field :

A vertical plane passing through the geographic axis is called geographic meridian and a great circle perpendicular to Earth's geographic axis is called geographic equator.
A vertical plane passing through magnetic axis is called magnetic meridian and a great circle perpendicular to Earth's magnetic axis is called magnetic equator.
The horizontal component of earth's magnetic field ( $B_{H}$ ) -It is the component of the earth's total magnetic field $B$ in the horizontal direction in the magnetic meridian.
The vertical component of earth's magnetic field ( $B_{V}$ ) -It is the component of the earth's total magnetic field $B$ in the vertical direction in the magnetic meridian.
The magnetic declination -The angle between the geographic meridian and the magnetic meridian at a place is called the magnetic declination at that place.
The angle subtended by the Earth's total magnetic field $B$ with the horizontal direction in the magnetic meridian is called dip or magnetic inclination (I) at that point.

H o r i z o n t a l C o m p o n e n t : B_{H} = B c o s I

V e r t i c a l C o m p o n e n t : B_{V} = B s i n I

t a n I = \frac{B _{V}}{B _{H}}

B = B_{H}^{2} + B_{V}^{2}

6. Important terms associated with Magnetism :

Magnetisation:

Each electron in an atom has a magnetic dipole moment associated with it. In a bulk medium, net magnetic moment arises from vector addition of moments of all atoms. Magnetisation is defined as magnetic moment per unit volume

M = \frac{m _{n e t}}{V}

Units:

A / m

Magnetising Field

The magnetic field which is used to magnetize a sample or specimen is called the magnetising field. Magnetising field is a vector quantity and is denoted by

H

and its unit is the same as that of magnetisation,

A / m

Magnetic Permeability

The magnetic permeability is the measure of ability of a material to allow the passage of magnetic field lines

through it. In free space, the permeability (or absolute permeability) is denoted by

μ_{o}

and for any other medium it is denoted by

μ

.The relative permeability

μ_{r}

is defined as the ratio between absolute permeability of the medium to the permeability of free space.

μ_{r} = \frac{μ}{μ _{o}}

Magnetic induction

The total magnetic induction (total magnetic field) inside the specimen

B

is equal to the sum of the magnetic field

B_{o}

produced in vacuum due to the magnetising field and the magnetic field

B_{m}

due to the induced magnetism of the substance.

B = B_{o} + B_{m}

B = μ_{o} (H + M)

Magnetic susceptibility

Magnetic susceptibility measures how easily and how strongly a material can be magnetised. It is defined as the ratio of the intensity of magnetisation(

M

) induced in the material to the magnetising field

H

. It is a dimensionless quantity

χ = \frac{∣ M ∣}{∣ H ∣}

7. Classification of Magnetic materials :

Diamagnetic substances

Magnetic susceptibility is negative.
Relative permeability is slightly less than unity.
The magnetic field lines are repelled or expelled by diamagnetic materials when placed in a magnetic field.
When placed in a non-uniform magnetic field, diamagnetic materials from stronger part to weaker part of the external field.
Susceptibility is nearly temperature independent.

Paramagnetic substances

Magnetic susceptibility is positive and small.
Relative permeability is greater than unity.
The magnetic field lines are attracted into the paramagnetic materials when placed in a magnetic field.
When placed in a non-uniform magnetic field, the paramagnetic materials will have a tendency to move from weaker to stronger part of the field.
Susceptibility is inversely proportional to temperature.

Curie's Law

When temperature is increased, thermal vibration upsets the alignment of magnetic dipole moments. Therefore, the magnetic susceptibility decreases with increase in temperature.

χ = \frac{C}{T}

where C is Curie's temperature

Ferromagnetic substances

Magnetic susceptibility is positive and large.
Relative permeability is large.
The magnetic field lines are strongly attracted into the ferromagnetic materials when placed in a magnetic field.
When placed in a non-uniform magnetic field, the ferromagnetic materials will have a strong tendency to move from weaker to stronger part of the field.
Susceptibility is inversely proportional to temperature.

Curie-Weiss law

As temperature increases, the ferromagnetism decreases due to the increased thermal agitation of the atomic dipoles. At a particular temperature, ferromagnetic material becomes paramagnetic. This temperature is known

as Curie temperature

T_{C}

χ = \frac{C}{T - T _{C}}

8. Hysteresis :

The word hysteresis means lagging behind. The phenomenon of lagging of intensity of magnetization (M) behind magnetic intensity (H), when a specimen of magnetic material is subjected to a cycle of magnetization is called hysteresis.

Retentivity

It is also known as residual magnetism or remanence which is the magnetic induction left behind in the sample after the magnetizing field has been removed.

Coercivity

Coercivity is defined as the minimum value of magnetising intensity that is required to bring the material to its original state.

Permanent Magnets and Electromagnets

Substances which at room temperature retain their ferromagnetic property for a long period of time are called permanent magnets. The material should have high retentivity so that the magnet is strong and high coercivity so that the magnetisation is not erased by stray magnetic fields, temperature fluctuations or minor mechanical damage.

Core of electromagnets are made of ferromagnetic materials which have high permeability and low retentivity. Soft iron is a suitable material for electromagnets.

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