Magnetism and Matter

Bar Magnet

A system composed of two poles, equal in magnitude but opposite in polarity, placed at a small displacement apart is known as a bar magnet. It is also known as a magnetic dipole.

Pole strength (m_p)

(Ia) The strength of a magnetic pole to attract magnetic materials towards it, is known as pole strength.

(b) Greater the number of unit poles in a magnetic pole, greater will be its strength.

(c) $m_p=\frac{F}{B}=\frac{Magneticforce}{Magneticinduction}$

(d) Its unit is ampere-meter.

Illustration 1: The magnetic flux density of magnetic field is 1.5 Wb/m². A proton enters this field with a velocity of 2 ×10⁷ m/s in a direction making an angle of 30° with the field. The force acting on the proton will be :

(A) 2.4 × 10^–12 Newton (B) 0.24 × 10^–12 Newton

(C) 24 x 10^–12 Newton (D) 0.024 × 10^–12 Newton

Solution : (A) $F=qvB\sin \theta =1.6\times {10}^{-19}\times 2\times {10}^7\times 1.5\times 0.5=2.4\times {10}^{-12}$

Magnetic lines of force

(a) The imaginary lines which represent the direction of magnetic field, are known as magnetic lines of force.

(b) The imaginary path traced by an isolated (imaginary) unit north pole is defined as a line of force.

(c) Magnetic lines of force are closed curves. Outside the magnet their direction is from north pole to south pole and inside the magnet these are from south to north pole.

(d) They neither have origin nor end.

(e) These lines do not intersect, because if they do so then it would mean two value of magnetic field at a single point, which is not possible.

(f) The tangent drawn at any point to the line of force indicates the direction of magnetic field at that point.

(g) At the poles of the magnet the magnetic field is stronger because the lines of force there are crowded together and away from the poles the magnetic field is week. i.e. magnetic field intensity $\alpha$ number of lines of force.

(h) The number of magnetic lines of force passing through unit normal area is defined as magnetic induction (B) whereas the number of lines of force passing through any area is known as magnetic flux.

(i) The lines of force can emerge out of the north pole of magnet at any angle and these can merge into the south pole at any angle.

Magnetic field

(a) The space around a magnet in which a torque acts on a magnetic needle is known as magnetic field.

(b) The space around a magnet in which a net force acts on a magnetic test pole is known as magnetic field.

(c) The space around a magnet in which its effect is experienced is known as magnetic field.

(d) There are four types of magnetic field :

(i) Uniform magnetic field: (a) The magnetic field, in which the intensity of magnetic field is same at all points, is known as uniform magnetic field.

(b) In such a magnetic field the magnetic lines of force are parallel and equidistant. e.g. the magnetic dines of force of earth's magnetic field.

(ii) Non-uniform magnetic field: (a) The magnetic field, in which the intensity of magnetic field at different points is different, is known as non-uniform magnetic field.

(b) It is represented by non-parallel lines of force.

(iii) Varying magnetic field: (a) The magnetic field, which keeps on changing with respect to time is known as a variable magnetic field.

(b) Example :– B = B₀ sin $\omega$t or B = B₀ cos $\omega$t

(iv) Non-varying magnetic field: (a) The magnetic field which does not change with time is known as a constant magnetic field.

(b) The direction of magnetic field is that in which a force acts on a unit test pole.

(c) It can be produced by moving charges, current carrying loops, and variations in electric currents.

Magnetic dipole

(a) The structures, which tend to align along the direction of magnetic field, are known as magnetic dipoles.

(b) Bar magnet, current carrying solenoid, current carrying coil, current carrying coil, current loop, magnetic needle etc. are the examples of magnetic dipole.

(c) It is not possible to separate out the two poles of a magnet.

(d) Magnetic dipole is not a system composed of two poles because the existence of monopoles is not possible.

A loop of single turn is also a magnetic dipole. One face of the loop behaves as north pole and the other face behaves as south pole. The face of the coil, in which current is anticlockwise, behaves as north pole and the face in which current is clockwise, behaves as south pole.

Magnetising field or intensity of magnetic field H

(a) The ratio of magnetic induction produced in vacum (B^o) and magnetic permeability of vacuum is defined as magnetising field (H), i.e. $H=\frac{B_{{}^o}}{{\mu }_o}$

(b) The intensity of magnetic field due to a pole of strength m_p at a distance r from it is $H=\frac{m_p}{\mu r^2}$

(c) Due to a small magnet H = $\frac{M}{r^3}\sqrt{1+3{\cos }^2\theta }$

Illustration 2: 1000 turns per meter are wound over a Rowland ring of ferromagnetic material. On passing a current of 2 ampere in the coil, a magnetic field of 10 Wb/m² is produced in it. The magnetising force generated in the material will be :

(A) 1.2 × 10^–3 A/m (B) 2.6 × 10^–3 A/m

(C) 2.6 × 10^–4 A/m (D) 2 × 10³ A/m

Solution: (D) $H=\frac{Ni}{2\pi r}=ni={10}^3\times 2=2\times {10}^{3}A/m$

Illustration 3: In the above problem, the value of intensity of magnetisation in A/m will be :

(A) 7.96 × 10⁶ (B) 7.96 × 10^–6

(C) 3.98 × 10³ (D) zero

Solution: (A) $B={\mu }_0\left(\mathrm{I}+H\right)$

$\therefore \mathrm{I}=\frac{B}{{\mu }_0}-H=\frac{10}{4\pi \times {10}^{-7}}-2\times 10^3=7.96\times 10^{6}A/m$

Illustration 4: In the above question 115 the relative permeability of the material will be :

(A) 4.98 × 10³ (B) 4.98 ×10^–3

(C) 2.98 × 10^–3 (D) 3.98 × 10³

Solution: (D) ${\mu }_r=1+X=1+\frac{\mathrm{I}}{H}=1+\frac{7.96\times {10}^6}{2\times {10}^3}=3.98\times 10^3$

Magnetic moment (M)

(a) If a magnet of length l and magnetic moment M is bent in the form of a semicircular are then its new magnetic moment will be M' = $\frac{2M}{\pi }$

(b) The magnetic moment of an electron due to its orbital motion is 1$\mu$_B whereas that due to its spin motion it is $\frac{{\mu }_B}{2}$.

i.e. M_orbital = $l\left(\frac{eh}{4\pi m_l}\right)=\left(\frac{\bar{e}h}{2m_l}\right)$

and M_spin = s $\left(\frac{eh}{4\pi m_s}\right)=s\left(\frac{\bar{e}h}{2m_s}\right)=s{\mu }_B=\frac{{\mu }_B}{2}$

Here $\mu$_B = Bohr magneton

(i) The value of Bohr magneton $\mu$B = $\frac{eh}{4\pi m}$

(ii) $\mu$B = 0.93 × 10^–23 Amp-m²

(c) Other formulae of M:

(i) M = ni$\pi$r²

(ii) $M=\frac{eVr}{2}=\frac{e{r}^2\omega }{2}=\frac{e{r}^{2}2\pi f}{2}=\frac{e{r}^2\pi }{T}$

(iii) $\frac{e}{2mJ}$

(iv) M = n$\mu$_B

(d) Resultant magnetic moment :

(i) When two bar magnets are lying mutually perpendicular to each other, then

$M=\sqrt{M_1^2+M_2^2}=\sqrt{2}{m}_{p}l$

(ii) When two coils, each of radius r and carrying current i, are lying concentrically with their planes at right angles to each other, then

$M=\sqrt{M_1^2+M_2^2}=\sqrt{2i}\pi r^2$ if M₁ = M₂

Illustration 5: A square loop OABCO of side $\ell$ carries a current i. It is placed as shown in figure. Find the magnetic moment of the loop.

Solution: Magnetic moment of the loop can be written as,

$\overrightarrow{M}=i\left(\overrightarrow{BC}\times \overrightarrow{CO}\right)$

Here, $\overrightarrow{BC}=-\ell \stackrel{\wedge }{k}$

$\overrightarrow{CO}=-\ell \cos {60}^{\circ }\stackrel{\wedge }{i}-\ell \sin {60}^{\circ }\stackrel{\wedge }{j}=-\frac{\ell }{2}\stackrel{\wedge }{i}-\frac{\sqrt{3}}{2}\ell \stackrel{\wedge }{j}$

$\therefore \overrightarrow{M}=i\left[(-\ell \stackrel{\wedge }{k})\times \left(-\frac{\ell }{2}\stackrel{\wedge }{i}-\frac{\sqrt{3}\ell }{2}\stackrel{\wedge }{j}\right)\right]$

or $\overrightarrow{M}=\frac{i{\ell }^2}{2}\left(\stackrel{\wedge }{j}-\sqrt{3}\stackrel{\wedge }{i}\right)$ Ans.

Intensity of magnetisation (I)

(a) The magnetic moment per unit volume of a material is defined as intensity of magnetisation (I). i.e. I = $\frac{M}{V}$, when

V = 1m³ then I = M.

(b) The unit of intensity of magnetisation is ampere/meter and its dimensions are M⁰L^–1T⁰A¹

(c) The pole strength per unit area of cross-section is defined as intensity of magnetisation. i.e. $I=\frac{m_p}{A}$when A = 1m² then A=1m² then I = mp

(d) It is a vector quantity whose direction is along the magnetic field.

(e) In para- and ferro-magnetic materials its direction is in the direction of H and in dia-magnetic materials it is opposite to that of H.

(f) I is produced in materials due to spin motion of electrons.

(g) The value of I and its direction in a material depend on the nature of that material.

(h) I - M curve

(i) For para magnetic materials

(ii) For diamagnetic materials

(iii) For ferromagnetic materials

(i) I - H curve

(j) Its value depends on temperature.

(k) It is produced on account of induction in a material.

(l) For low magnetising field I $\propto$ H. i.e. I $\propto$ H or I = $\chi$Ht

(m) $\chi$ is a dimensionless constant i.e. it carries no unit.

(n) Other types of intensity of magnetisation:

(i) Mass intensity of magnetisation (I_mass)-

$I_{mass}or{I}_m=\frac{Intensityofmagnetisation}{densityofmagneticmaterial}=\frac{1}{\rho }$

(ii) Molar intensity of magnetisation-

I_molar = I_MW = Mo. Wt.× mass intensity of magnetisation = M_W × I_m

(iii) Molecular intensity of magnetisation-

$I_{molecular}=I_M=\frac{Molar\mathrm{int}ensityofmagnetisation}{AvogadroNo.}=\frac{I_{MW}}{N}$

Magnetic susceptibility or ($\chi$ or K)

(a) The ratio intensity of magnetisation (I) in a material and the magnetising field (H) is defined as magnetic susceptibility ($\chi$).

i.e. $\chi =\frac{I}{H}$

When H = 1 Oersted then $\chi$ = I.

(b) The intensity of magnetisation induced in a material by unit magnetising field is defined as magnetic susceptibility.

(c) It has no unit and no dimensions.

(d) It is a measure of ease with which a material can be magnetised by a magnetised by a magnetising field (H).

(e) Magnetic susceptibility of various materials-

(i) For diamagnetic materials – $\chi$ = low and negative

(ii) For paramagnetic materials – $\chi$ = low but positive

(iii) For ferromagnetic materials – $\chi$ = high and positive

(f) For paramagnetic substances it is inversely proportional to temperature i.e. $\chi \propto \frac{I}{T}$

(g) For low magnetising field the value of $\chi$ is constant.

(h) Different types of magnetic susceptibility-

(i) Volume susceptibility

${\chi }_V=\frac{I}{H}=X$

(ii) Mass or specific susceptibility $\chi$_m

${\chi }_m=\frac{{\chi }_v}{\rho }=\frac{Volumesusceptibility}{densityofmeterial}$

(iii) Molar susceptibility $\chi$_MW

$\chi$_MW = $\chi$_m x M_W

$\chi$ = Specific susceptibility × Mol. Wt.

(iv) Molecular susceptibility $\chi$_m

$\chi$m = Atomic wt. x specific susceptibility

= A ${\chi }_m=\frac{AI}{\rho H}$

(i) $\chi$–T curve

(j) $\chi$–$\frac{1}{T}$ :

Illustration 6: The magnetic induction along the axis of an air solenoid is 0.03 Tesla. On placing an iron core inside the solenoid the magnetic induction becomes 15 Tesla. The relative permeability of iron core will be :

(A) 300 (B) 500

(C) 700 (D) 900

Solution: (B) $B_m=\mu H,B={\mu }_{0}H\therefore {\mu }_r=\frac{\mu }{{\mu }_0}=\frac{B_m}{B}=\frac{15}{0.03}=500$

Absolute magnetic permeability ($\mu$)

(a) The ratio of magnetic induction (B) to magnetising field (H) is defined as magnetic permeability ($\mu$).

(b) The extent to which magnetic permeability of that medium.

(c) It is the characteristic property of a magnetic material because it represents the amplification of magnetising field in that material.

(d) Its value is always positive and is different for different materials.

(e) For materials its value can be greater or less than $\mu$₀.

(f) Its value depends on H and T.

(h) $\mu$ = $\mu$₀ [1 + $\chi$]

(i) $\mu$ = $\mu$₀$\mu$_r

(j) (i) For feeromagnetic materials $\mu$ = high

(ii) For paramagnetic materials $\mu$ = low

(iii) For diamagnetic materials $\mu$ = very low

(k) The unit of magnetic permeability is Weber per ammere-meter or Henry per meter and its dimensions are M¹L⁰T^–2A^–2.

Relative permeability ($\mu$_r)

(a) The ratio of magnetic permeability of medium ($\mu$) to the magnetic permeabiliy of free space ($\mu$₀) is defined as relative permeability ($\mu$_r). i.e. ${\mu }_r=\frac{\mu }{{\mu }_0}$

(b) ${\mu }_r=\frac{Numberoflinesofforcepas\sin gthroughunitareainmedium}{Numberoflinesofforcepas\sin gthroughunitareainvacuum}$

(c) ${\mu }_r=\frac{magneticfluxdensityinmedium}{Magneticfluxdensityinvacuum}$

(d) The limit unto which a magnetic field penetrates matter, is known as relative permeability of that material.

(e) It has no unit and no dimensions.

(f) $\mu$_r = 1 + $\chi$

(g) Relative permeability of various substances-

(i) For diamagnetic substances the value of $\mu$_r is slightly less than one i.e. $\mu$_r < 1.

(ii) For paramagnetic substances the value of $\mu$_r is slightly greater than one i.e. $\mu$_r > 1.

(iii) For ferromagnetic substance the value of $\mu$_r is much greater than one i.e. $\mu$_r >> 1.

MAGNETIC DIPOLE IN A UNIFORM MAGNETIC FIELD

(i) When a dipole is placed (or suspended) in a uniform magnetic field, then no net force acts on it i.e. the resultant force acting on it is zero.

(ii) When a dipole is placed in a uniform magnetic field then a torque acts on it which is given by

$\tau =\vec{M}\times \vec{B}$ Newton-meter

(iii) The magnitude of torque is given by

$\tau =MB\sin \theta =2mlB\sin \theta$

Where m is the pole strength and 2l is the length of dipole.

(iv) The units of M are ampere-meter² or Joule/Tesla and its dimensions are M⁰L²A¹

(v) When a magnetic dipole is placed in a non-uniform magnetic field, then a torque as well as a force both act on it. The force acting on the dipole is given by

$\vec{F}=\vec{M}\times \frac{\overrightarrow{\mathrm{dB}}}{dr}Newton$.

(vi) (a) The work done in rotating a dipole in a magnetic field from an angle $\theta$₁ to angle $\theta$₂ with the field direction is given by-

W = MB (cos $\theta$₁ – cos $\theta$₂) Joule

(b) If initially the dipole is lying along the field direction, then $\theta$₁ = 0

$\therefore$ W = MB (1 – cos $\theta$)

(vii) Potential energy (U): (a) The potential energy of a dipole in a magnetic field is given by

U = $\vec{M}.\vec{B}Joule$

(b) Special cases:

When $\theta$ = 0, then

U = – MB

When $\theta$ = 90^o, U = 0

When $\theta$ = 180^o, U = – MB (–1) = MB

Illustration 7: A Magnet is suspended in the magnetic meridian with a untwisted wire. The upper end of the wire is rotated through 180° to deflect the magnet by 30° from magnetic meridian. Now this magnet is replaced by another magnet. Now the upper end of the wire is rotated through 270° to deflect the magnet by 30° from magnetic meridian. Compare the magnetic moments of magnets.

Solution: If $\varphi$be the twist of the wire, then $\tau =C\varphi$, where C being restoring couple per unit twist of wire. Here

$\varphi$₁ = 180° – 30° = 150° = (150 x $\pi$/150) radian

$\varphi$₂ = (270° – 30° )= 240° = (240 x $\pi$/180) radian

If M be the magnetic moment and H, the horizontal component of earth's field, then $\tau =C\varphi$, = MH sin$\theta$

$\therefore C\varphi$ = MH sin$\theta$

If M₁ and M₂ be the magnetic moments of the two magnets respectively, then

$\therefore C\varphi$₁ = M₁H sin$\theta$ for first magnet

$\therefore C\varphi$₂ = M₂H sin$\theta$ for second magnet

$\therefore \frac{{\varphi }_1}{{\varphi }_2}=\frac{M_1}{M_2}or\frac{M_1}{M_2}=\frac{150\times \left(\pi /180\right)}{240\times \left(\pi /180\right)}=\frac{5}{8}$

$\therefore$ M₁ : M₂ = 5 : 8

Classification of materials

(i) The root cause of magnetism in matter is the motion of electric charges.

(ii) The motion of electrons and protons in atoms is responsible for their magnetic properties.

(iii) The variation in the number of fundamental charged particles and variation in their arrangement in different materials are responsible for differences in their magnetic properties.

(iv) On the basis of mutual interactions or behaviour of various materials in an external magnetic field, the materials are divided in three main categories.

(1) Diamagnetic substances (2) Paramagnetic substances

(3) Ferromagnetic substances

Comparative study of these materials:

Curie law and Curie temperature

(a) Curie law: (i) The magnetic susceptibility of paramagnetic substances in inversely proportional to its absolute temperature i.e. $\chi \propto \frac{1}{T}$

(ii) $\chi \propto \frac{C}{T}$Where C = Curie constant, T = absolute temperature

(iii) On increasing temperature, the magnetic susceptibility of paramagnetic materials decreases and vice versa.

(iv) The magnetic susceptibility of ferromagnetic substances does not change according to curie law.

(b) Curie temperature (T_C): (i) The temperature above which a ferromagnetic material behaves like a paramagnetic material is defined as curie temperature (T_C).

(ii) The minimum temperature at which a ferromagnetic substance is converted into paramagnetic substance is defined as curie temperature.

(iii) For various ferromagnetic materials its values are different. e.g. for Ni $T_{C_{Ni}}=358{}^{\circ }C$

for Fe $T_{C_{Fe}}=770{}^{\circ }C$

for CO $T_{C_{CO}}=1120{}^{\circ }C$

(iv) At this temperature the ferromagnetism of the substances suddenly vanishes.

Curie-Weiss law

$\chi$–T Cure

Other important features and formulae

(A) Angle of declination and Geographical meridian:

(i) The horizontal component of earth's magnetic field is from S to N.

(ii) If at any place the angle of dip is $\theta$ and magnetic latitude is$\lambda$then,

tan $\theta$ = 2 tan $\lambda$

The total intensity of earth's magnetic field

$\mathrm{I}={\mathrm{I}}_0\sqrt{1+3{\sin }^2\lambda }$

here ${\mathrm{I}}_0=\frac{M}{R^3}$

Here M and R are the magnetic moment of bar magnet of earth and radius of earth respectively.

(iii) At magnetic equator of earth$\lambda$= 0° and at poles$\lambda$= 90°.

${\mathrm{I}}_{pole}=2{\mathrm{I}}_{equator}$

(iv) At the poles and equator of earth, the values of total intensity are 0.66 and 0.33 oersted respectively.

(B) (i) In vacuum :– B₀ = $\mu$₀H

(ii) In medium :– B = $\mu$H

(iii) Resultant magnetic field due to Iand H :–

(a) B = B₁ + BH

= $\mu$₀I + $\mu$₀H

B = $\mu$₀ (I + H)

B = $\mu$₀H (1 + I/H)

or B = $\mu$₀ (1 + $\chi$)H + = $\mu$H

(iv) (a) $\mu$ = $\mu$₀ [1 + $\chi$]

(b) $\mu$ = $\mu$₀$\mu$_r

(c) $\mu =\frac{B}{H}$

(v) (a) ${\mu }_r=\frac{\mu }{{\mu }_0}$

(b) ${\mu }_r=1+\chi$

(vi) (a) $\chi ={\mu }_r-1$

(b) $\chi =\frac{\mathrm{I}}{H}$

(vii) The magnetic potential due to a small magnet at a point distant r is given by :–

$V=\frac{M\cos \theta }{r^2}$

(viii) The mutual interaction force between two small magnets of moments M₁ and M₂ is given by

$F=K\frac{6M_1{M}_2}{r^4}$

(C) Magnetic torque t

(a) $\tau$ = MB sin $\theta$

(b) $\tau$ = BiNA sin $\theta$

(c) $\tau =\vec{M}\times \vec{B}$

(D) Magnetic potential energy (U_B)

(a) $U_B=\vec{M}\cdot \vec{B}$

(b) $U_B=M\cdot B\left(1-\cos \theta \right)$