Magnetic Properties: theory

CONCEPT

Bar Magnet & Magnetic Moment

In one line
A magnet is a dipole: two equal and opposite poles a small distance apart, with moment M from S to N.

MOLECULAR THEORY

22.1.1 What is inside a magnet

Weber and Ewing: every molecule is a tiny magnet.
Unmagnetised: molecular magnets are random, net moment zero.
Magnetised: they line up, so a net moment appears.

Molecular magnets

BAR MAGNET

22.1.2 Effective and geometric length

Poles are not exactly at the ends.
Effective length is the distance between the poles.

L_e = 2l = 56 L_g

Semicircular magnet: L_g = πR, L_e = 2R.

Effective and geometric lengths

PROPERTIES

22.1.3 Properties of a magnet

Directive: a freely suspended magnet rests in the N-S direction (magnetic meridian).

Freely suspended magnet

No monopole: break a magnet and each piece has two poles.

Each piece is a complete magnet

Field lines are closed loops: S to N inside, N to S outside.
Repulsion is the sure test of magnetism: attraction also happens with plain iron.

Lines form closed loops

Imaginary isolated poles

Only Q is a magnet

POLE STRENGTH

22.1.4 Pole strength (m)

Scalar. N pole +m, S pole −m.
SI unit: A m. Dimensions: [LA].
Depends on material and area of cross-section.
Does not depend on length.

More area, more pole strength

MAGNETIC MOMENT

22.1.5 Magnetic moment (M)

M = m(2l)

Vector from S to N.
SI unit: A m², also J/T. Dimensions: [AL²].

Direction of M

CURRENT LOOP

22.1.6 Current loop as a magnet

M = NIA

Direction: curl fingers along I; thumb gives M.
Solenoid = many dipoles in a line.
Toroid and straight wire: M = 0.
Atoms with all electrons paired: M = 0.

Moment of a loop

Solenoid behaves as a bar magnet

A wire of length L bent into one circle:

M = IL²4π

Charge q moving in a circle of radius R at speed v:

M = qvR2

Electron: L and M are opposite, angle 180°.
Bohr magneton is the natural unit:

μ_B = eh4πm = 9.27 × 10⁻²⁴ A m²

CUTTING

22.1.7 Cutting a magnet

Cut into n parts	m′	M′
across the length	m	Mn
along the length	mn	Mn

BENDING

22.1.8 Bending a magnet

m is unchanged. Use the new end-to-end distance.

M′ = M × end-to-end distancelength

New length is the chord

Shape	M′
semicircle	2Mπ
arc of 60°	3Mπ
bent at the middle at 60°	M2
bent at the middle at 90°	M√2

JEE · NEET

Corner

Cutting a rectangular magnet into n equal parts

Cut along and across the length at the same time

Applies when cut both along and across, n parts in all.

L′ = L√n · b′ = b√n · w′ = wn

m′ = m√n · M′ = Mn

I = w L² + b²12 · I′ = In²

Cutting a thin magnet across its length

L′ = Ln · w′ = wn · M′ = Mn · I′ = In³

Practise PYQs on this topicBar Magnet & Magnetic Moment · 15 questions with solutions

CONCEPT

B, H, I, χ and μ: The Five Quantities

In one line
The current sets H, the material answers with I, and together they give the total field B.

How the quantities link up

FLUX AND B

22.2.1 Flux and magnetic induction (B)

Flux φ: number of lines through a surface. SI: weber. CGS: maxwell.
1 Wb = 10⁸ maxwell.
B is flux per unit normal area. It is a vector.

B = φA

SI: tesla (Wb/m²). CGS: gauss.
1 T = 10⁴ G.

Flux through area A

PERMEABILITY

22.2.2 Permeability (μ)

How easily lines pass through a material.

μ = μ₀μ_r · μ_r = BB₀

μ₀ = 4π × 10⁻⁷ T m/A.
μ_r has no unit.

Lines crowd into soft iron

MAGNETISING FIELD

22.2.3 Magnetising field (H)

H = Bμ

SI: A/m. CGS: oersted.
1 oersted = 80 A/m.

ROD IN A FIELD

22.2.4 Iron rod in a field

Rod in H, flux φ, area A. Convert cm² to m² first.

μ = BH = φAH

MAGNETISATION

22.2.5 Intensity of magnetisation (I)

Moment per unit volume, or pole strength per unit area.
Vector. SI unit: A/m.

I = MV = mA

Volume from mass: V = massdensity.
Volume of a bar: V = length × area.
N atoms, moment μ each: total M = Nμ.

SUSCEPTIBILITY

22.2.6 Susceptibility (χ)

χ = IH

Scalar with no unit.
How easily the material is magnetised.

RELATIONS

22.2.7 Relations between them

B = μ₀(H + I)

B = μ₀H(1 + χ)

μ_r = 1 + χ

μ = μ₀(1 + χ)

Note: in the question, I may be printed as M. Read which one it means: moment or magnetisation.
μ₀4π = 10⁻⁷ SI.

UNITS

22.2.8 Units

Quantity	SI unit	Type
φ	Wb	scalar
B	T = Wb/m²	vector
H	A/m	vector
I	A/m	vector
M	A m²	vector
μ	T m/A = N/A²	scalar
χ, μ_r	none	scalar

Practise PYQs on this topicMagnetisation: B, H, M, χ & μ · 45 questions with solutions

CONCEPT

Solenoid & Toroid with a Core

In one line
A core multiplies the field of a solenoid by μ_r; the increase is simply χ as a fraction.

SOLENOID

22.3.1 Solenoid

n = turns per metre, i = current.
H depends only on the current, not on the core.

H = ni

B₀ = μ₀ni

B = μ₀μ_rni = μ₀ni(1 + χ)

Core inside the winding

N turns over length L: n = NL. Convert cm to m.
Diameter of the tube does not enter.

TOROID

22.3.2 Toroid

n = N2πr

Toroid of mean radius r

Field lines stay inside the toroid; there are no poles.

MAGNETISATION OF THE CORE

22.3.3 Magnetisation of the core

I = χH = (μ_r − 1) ni

When μ_r is large, I ≈ μ_rni.

PERCENTAGE INCREASE

22.3.4 Percentage increase in B

B − B₀B₀ = χ

% increase = χ × 100

Fractional increase: χ itself.
Percentage: multiply by 100.
Do not add 1: the answer is not (1 + χ) × 100.

ROD WITH A WINDING

22.3.5 Rod with a winding

Step 1: H = ni.
Step 2: B = φA.
Step 3:

μ_r = Bμ₀H

Practise PYQs on this topicMagnetisation: B, H, M, χ & μ · 45 questions with solutions

CONCEPT

Field of a Magnet & Magnet in a Uniform Field

In one line
A magnet in a uniform field feels no net force, only a torque that turns it towards the field.

FIELD OF A SHORT MAGNET

22.4.1 Field of a short magnet

Axial and equatorial points

axial: B_a = μ₀4π 2Mr³

equatorial: B_e = μ₀4π Mr³

B_a : B_e = 2 : 1 at equal distance.
B_a is along M; B_e is opposite to M.
Both vary as 1r³.
Two magnets at right angles: add the fields as vectors.

TORQUE

22.4.2 Torque (τ)

Forces mB on the poles form a couple

τ = MB sin θ

τ = M × B

Each pole feels F = mB.
Uniform field: net force is zero, torque may not be.
Non-uniform field: both force and torque.
A current loop behaves the same, with M = NIA.

θ is the angle between M and B. Plane of a coil at α to B means θ = 90° − α.

WORK

22.4.3 Work done

W = MB(cos θ₁ − cos θ₂)

From the field direction, θ₁ = 0°:

W = MB(1 − cos θ)

POTENTIAL ENERGY

22.4.4 Potential energy (U)

U = −MB cos θ = −M · B

θ	τ	U	W from 0°	State
0°	0	−MB	0	stable
60°	√32 MB	−MB2	MB2
90°	MB	0	MB
180°	0	+MB	2MB	unstable

Two positions of equilibrium

W AND τ TOGETHER

22.4.5 W and τ together

Work W to turn through θ. Torque to hold it there:

τ = W sin θ1 − cos θ

θ = 60°: τ = √3 W.
θ = 30°: τ = W2 − √3.
Given τ at angle θ: find MB, then 2MB or U.

JEE · NEET

Corner

Coulomb’s law of magnetism

F = μ₀4π m₁m₂r²

μ₀4π = 10⁻⁷ Wb per A m.
In CGS, the constant is 1.

Exact field of a bar magnet of length 2l

B_a = μ₀4π 2Mr(r² − l²)²

B_e = μ₀4π M(r² + l²)^1.5

General position and force between dipoles

B_g = μ₀4π Mr³ √(3cos²θ + 1)

Force between two short magnets, end-on: F ∝ 1x⁴.
Gauss’s law: net flux through any closed surface is zero, ∮B · dS = 0.

Practise PYQs on this topicField of a Magnet & Magnet in a Uniform Field · 33 questions with solutions

CONCEPT

Earth’s Magnetism

In one line
The earth behaves like a huge magnet with three elements: declination, dip and B_H.

THE EARTH AS A MAGNET

22.5.1 The earth as a magnet

Magnetic and geographic axes

Likely cause: currents in the molten core.
Magnetic meridian: vertical plane through the magnetic axis.
The field changes from place to place and with time.

THREE ELEMENTS

22.5.2 Three elements

Declination θ: angle between geographic and magnetic meridians.

Declination

Dip δ: angle of B with the horizontal in the magnetic meridian.
B_H: horizontal component of the field.

Dip and the two components

B_H = B cos δ

B_V = B sin δ

B = √(B_H² + B_V²) · tan δ = B_VB_H

POLES AND EQUATOR

22.5.3 Poles and equator

Place	δ	B_H	B_V
magnetic equator	0°	B	0
magnetic pole	90°	0	B

At a pole, a horizontal compass needle can stay in any direction.
Dip is positive in the northern hemisphere, negative in the southern.

JEE · NEET

Corner

Apparent dip

Plane at angle β to the magnetic meridian

In another vertical plane, B_V stays the same; B_H becomes B_H cos β.

tan δ′ = tan δcos β

Apparent dip is greater than or equal to true dip.
Two vertical planes at right angles:

cot²δ = cot²δ₁ + cot²δ₂

Earth as a dipole

B_equator = μ₀4π MR³

Magnetic axis makes about 11.5° with the geographic axis.
At magnetic latitude λ:

tan δ = 2 tan λ

Southern half of the earth’s magnet lies in the northern hemisphere.
Total intensity: pole 0.66 oersted, equator 0.33 oersted.

Neutral points

Resultant field is zero: the magnet’s field cancels B_H.
N pole towards north: neutral points on the equatorial line.

μ₀4π Mr³ = B_H

N pole towards south: neutral points on the axial line.

μ₀4π 2Mr³ = B_H

Magnetic maps

Line	Joins places of equal
isogonic	declination
agonic	zero declination
isoclinic	dip
aclinic	zero dip (magnetic equator)
isodynamic	B_H

Tangent law and tangent galvanometer

Needle rests under B and B_H at right angles

B = B_H tan θ

Circular coil with a compass at its centre

Coil is kept in the magnetic meridian.

μ₀Ni2r = B_H tan θ

i = k tan θ · k = 2rB_Hμ₀N

k is the reduction factor. It measures current.

Deflection magnetometer

Works on the tangent law

Tan A: B_H tan θ = μ₀4π 2Mr³

Tan B: B_H tan θ = μ₀4π Mr³

M₁M₂ = tan θ₁tan θ₂ · null method: M₁M₂ = (d₁d₂)³

Practise PYQs on this topicEarth's Magnetism · 16 questions with solutions

CONCEPT

Vibration Magnetometer

In one line
A magnet displaced from the field direction performs S.H.M.: stronger M or B, faster oscillation.

TIME PERIOD

22.6.1 Time period

Magnet oscillating in B_H

T = 2π √(IMB)

ν = 12π √(MBI)

I = wL²12

Here I is moment of inertia, not magnetisation.
Small angle: τ = −MBθ gives S.H.M.
Frequency from a count: ν = oscillations per second.

FINDING M OR B

22.6.2 Finding M or B

M = 4π²IBT² · B = 4π²IMT²

COMPARING

22.6.3 Comparing fields and magnets

Same magnet: B₁B₂ = T₂²T₁² = ν₁²ν₂²

Same size, same B: M₁M₂ = T₂²T₁²

Same B: M₁M₂ = I₁I₂ × T₂²T₁²

An added field opposite to B_H gives net B = B_H − B′; along it, B_H + B′.
Field of a magnet at a new place: use the total B, not the increase.

SUM AND DIFFERENCE

22.6.4 Sum and difference positions

Like poles together and opposed

T_s = 2π √(I₁ + I₂(M₁ + M₂)B)

T_d = 2π √(I₁ + I₂(M₁ − M₂)B)

M₁M₂ = T_d² + T_s²T_d² − T_s²

Moment of inertia always adds.
T_d is greater than T_s.
Equal moments, difference position: M_net = 0, so T_d = ∞.

CHANGES TO THE MAGNET

22.6.5 Changes to the magnet

Change	I	M	T′
mass × 4 (same size)	4I	M	2T
thin, cut across into n	In³	Mn	Tn
cut along and across, n parts	In²	Mn	T√n

Heating: M falls, so T rises.
At the Curie temperature it stops oscillating.

JEE · NEET

Corner

Compass needle at two places of different dip

Only B_H = B cos δ makes the needle oscillate.

ν₁²ν₂² = B₁ cos δ₁B₂ cos δ₂

Ratio of a magnet’s field to B_H

Secondary magnet placed near the primary

BB_H = (ν′ν)² − 1

Practise PYQs on this topicVibration Magnetometer · 20 questions with solutions

CONCEPT

Dia, Para & Ferromagnetic Materials

In one line
Materials differ by what the atoms do: dia opposes the field, para weakly follows it, ferro strongly follows it.

ORIGIN

22.7.1 Origin of magnetism

Diamagnetic: paired electrons, atom has no net moment. Moment is induced by orbital motion, opposite to the field. Every material has this property.
Paramagnetic: unpaired spins, each atom has a moment. Without a field the moments are random; a field lines them up a little.
Ferromagnetic: atoms align together in groups called domains.

Paramagnetic atoms

DOMAINS

22.7.2 Domains

A domain is a region fully magnetised to saturation.
Unmagnetised: domains point in different directions.
In a field: favourable domains grow, others shrink, and domains rotate.
Atoms interact and align by themselves.

Domains without and with a field

BEHAVIOUR

22.7.3 Behaviour in a field

Field lines near the three materials

Dia: lines bend away. Para: lines come in slightly. Ferro: lines crowd in strongly.
Rod in a uniform field: dia sets perpendicular; para and ferro set parallel.

Liquid in a U-tube with one arm between poles

COMPARISON

22.7.4 Comparison

Property	Dia	Para	Ferro
χ	small, negative	small, positive	large, positive
range	−1 ≤ χ < 0	0 < χ < ε	χ ≫ 1
μ_r	0 ≤ μ_r < 1	slightly above 1	μ_r ≫ 1
B inside	B < B₀	B > B₀	B ≫ B₀
I and H	opposite	same way	same way, large
magnet	weakly repelled	weakly attracted	strongly attracted
moves	strong to weak field	weak to strong	weak to strong
atom’s moment	zero	non-zero	non-zero
gas between poles	expands across field	expands along field	rapidly along field
examples	Cu, Ag, Au, Zn, Bi, Sb, NaCl, H₂O, air, diamond	Al, Mn, Pt, Na, CaCl₂, O₂	Fe, Co, Ni, Fe₃O₄

I against H

SUPERCONDUCTOR

22.7.5 Superconductor: a perfect diamagnet

χ = −1 and μ_r = 0.
Field inside is zero: lines are fully expelled.
The relative permittivity of any material stays above 1.

Practise PYQs on this topicMagnetic Materials & Curie Law · 46 questions with solutions

CONCEPT

Curie Law & Curie Temperature

In one line
Heat disturbs alignment: χ of para falls as T rises, ferro turns para at T_c, and dia does not care.

CURIE LAW

22.8.1 Curie law

χ = CT · χT = constant

I = C BT

C is the Curie constant; T is in kelvin.
Convert °C: add 273.
Magnetisation rises with B and falls with T.

χ₂χ₁ = T₁T₂ · I₂I₁ = B₂B₁ × T₁T₂

χ AGAINST T

22.8.2 χ against T

Dia is flat and negative; para falls smoothly; ferro turns para at T_c

χ against 1T for a paramagnet: straight line through the origin.
Dia: χ does not depend on T (except Bi at low temperature).

CURIE TEMPERATURE

22.8.3 Curie temperature (T_c)

Above T_c a ferromagnet becomes paramagnetic.
Heating makes the domains random; atoms keep their own moments.
It does not become diamagnetic.

Material	T_c
Ni	358°C
Fe	770°C
Co	1120°C

CURIE-WEISS LAW

22.8.4 Curie-Weiss law

χ = CT − T_c

Valid for T greater than T_c

Practise PYQs on this topicMagnetic Materials & Curie Law · 46 questions with solutions

CONCEPT

Hysteresis, Soft Iron vs Steel & Shielding

In one line
A ferromagnet remembers: B lags behind H, and the loop shape decides what the material is used for.

HYSTERESIS CURVE

22.9.1 Hysteresis curve

Only ferromagnetic materials show hysteresis.

O to A: first magnetisation. A C D E F G A: one full cycle

OA: first magnetisation up to saturation at A.
OC: B left at H = 0: retentivity (residual magnetism).
OD: reverse H that makes B zero: coercivity.
E: saturation in the reverse direction.
Point on the B axis = retentivity. Point on the H axis = coercivity.

DEMAGNETISING

22.9.2 Demagnetising current

Put the magnet inside a solenoid, with H opposite to its magnetisation.

H_c = ni · i = H_cn

ENERGY LOSS

22.9.3 Energy loss

Area of the loop is the energy lost per unit volume per cycle.

Energy loss = VAft

V: volume, A: loop area, f: frequency, t: time.

SOFT IRON VS STEEL

22.9.4 Soft iron vs steel

Narrow and wide loops

Property	Soft iron	Steel
loop area	small	large
energy loss	low	high
retentivity	low	high
coercivity	low	high
permeability	high	lower
magnetise and demagnetise	easily	not easily
used for	electromagnet, transformer, dynamo core, tape recorder	permanent magnet

Electromagnet: high μ, low retentivity, low coercivity.
Permanent magnet: high retentivity, high coercivity.
Transformer core: narrow loop, low loss.

MAGNETIC SHIELDING

22.9.5 Magnetic shielding

Put the instrument in a soft iron enclosure.
High μ draws the lines into the wall.
Field inside is nearly zero.
Copper, aluminium or plain insulation will not do it.

Lines go through the iron wall

Practise PYQs on this topicHysteresis, Soft Iron vs Steel & Shielding · 20 questions with solutions

EXAM-DAY CARD

Just see this before the exam

M = m(2l) = NIA

moment

B = μ₀(H + I)

total field

μ_r = 1 + χ

permeability

U = −MB cos θ

energy

T = 2π√(IMB)

oscillation

χ = CT

Curie law

FORMULA SHEET

Whole chapter on one page

Bar magnet

M = m(2l), L_e = 56 L_g

loop: M = NIA

semicircle: M′ = 2Mπ

cut across: m same, M′ = Mn

B, H, I, χ, μ

B = φA, μ = BH

I = MV, χ = IH

B = μ₀(H + I)

μ_r = 1 + χ

With a core

H = ni

B = μ₀ni(1 + χ)

I = (μ_r − 1)ni

% increase = χ × 100

Magnet in a field

B_a = μ₀4π 2Mr³, B_e = μ₀4π Mr³

τ = MB sin θ

W = MB(1 − cos θ)

U = −MB cos θ

Earth & oscillation

B_H = B cos δ, B_V = B sin δ

T = 2π√(IMB)

B₁B₂ = T₂²T₁²

M₁M₂ = T_d² + T_s²T_d² − T_s²

Materials

dia: χ < 0; para: small χ > 0

ferro: χ ≫ 1

χ = CT, χ = CT − T_c

i = H_cn

TOP MISTAKES

Avoid these = easy marks

MISTAKE ✗

Using χ + 1 as the % increase

Curie law with T in °C

Above T_c: ferro becomes dia

Electromagnet from steel

CORRECT ✓

% increase = χ × 100

T always in kelvin

Above T_c: ferro becomes para

Soft iron: low retentivity and coercivity

Magnetic Properties

Bar Magnet & Magnetic Moment

22.1.1 What is inside a magnet

22.1.2 Effective and geometric length

22.1.3 Properties of a magnet

22.1.4 Pole strength (m)

22.1.5 Magnetic moment (M)

22.1.6 Current loop as a magnet

22.1.7 Cutting a magnet

22.1.8 Bending a magnet

Corner

B, H, I, χ and μ: The Five Quantities

22.2.1 Flux and magnetic induction (B)

22.2.2 Permeability (μ)

22.2.3 Magnetising field (H)

22.2.4 Iron rod in a field

22.2.5 Intensity of magnetisation (I)

22.2.6 Susceptibility (χ)

22.2.7 Relations between them

22.2.8 Units

Solenoid & Toroid with a Core

22.3.1 Solenoid

22.3.2 Toroid

22.3.3 Magnetisation of the core

22.3.4 Percentage increase in B

22.3.5 Rod with a winding

Field of a Magnet & Magnet in a Uniform Field

22.4.1 Field of a short magnet

22.4.2 Torque (τ)

22.4.3 Work done

22.4.4 Potential energy (U)

22.4.5 W and τ together

Corner

Earth’s Magnetism

22.5.1 The earth as a magnet

22.5.2 Three elements

22.5.3 Poles and equator

Corner

Vibration Magnetometer

22.6.1 Time period

22.6.2 Finding M or B

22.6.3 Comparing fields and magnets

22.6.4 Sum and difference positions

22.6.5 Changes to the magnet

Corner

Dia, Para & Ferromagnetic Materials

22.7.1 Origin of magnetism

22.7.2 Domains

22.7.3 Behaviour in a field

22.7.4 Comparison

22.7.5 Superconductor: a perfect diamagnet

Curie Law & Curie Temperature

22.8.1 Curie law

22.8.2 χ against T

22.8.3 Curie temperature (Tc)

22.8.4 Curie-Weiss law

Hysteresis, Soft Iron vs Steel & Shielding

22.9.1 Hysteresis curve

22.9.2 Demagnetising current

22.9.3 Energy loss

22.9.4 Soft iron vs steel

22.9.5 Magnetic shielding

Just see this before the exam

Whole chapter on one page

Avoid these = easy marks

22.8.3 Curie temperature (T_c)