Magnetic Properties_BSC

aMagnetic Properties and FieldDomain Theory of FerromagnetismIn most of the substance the magnetic effects are weak. But there are some kind of substances which exhibit strong magnetic effects. The ferromagnetic substance consists of large number of microscopic region in which magnetic movements are aligned called ferromagnetic domains. Each domain behaves as a tiny magnet but in the absence of magnetic field, moments are randomly oriented so that the net magnetic field is zero. In the presence of field, magnetic movements are aligned and substance is magnetized. Even if the field is removed magnetism is retained within the surface.

Magnetic Dipole MomentQ. What do you mean by the magnetic dipole moment of current loop?We know, the torque on a current loop of area A$\overrightarrow{A}$ through which a current flows is given by𝜏=I(A×B)$$\overrightarrow{𝜏}=\mathrm{I}(\overrightarrow{\mathrm{A}}\times \overrightarrow{\mathrm{B}})$$Where B$\overrightarrow{B}$ is magnetic field intensity. The torque acting on an electric dipole having electric dipole moment P$\overrightarrow{P}$ in an electric field is given by,T=P×E$$\overrightarrow{\mathrm{T}}=\overrightarrow{\mathrm{P}}\times \overrightarrow{\mathrm{E}}$$Comparing equations (i) and (ii), we find that the current loop in a uniform magnetic field behaves similar to a electric dipole in a uniform electric field i.e., small current loop has the properties of a magnetic dipole.The vector IA=Pm$\overrightarrow{IA}={\overrightarrow{P}}_m$ is callid the magnetic moment of the current loop Hence if a current loop of magnetic moment P$\overrightarrow{\mathrm{P}}$ है is placed in a unifonm magnctic ficid B$\overrightarrow{B}$, the toryue acting on it if given byr=P=B$$r=\overrightarrow{P}=\overrightarrow{B}$$In other words, a current carrying loop of wire behaves as magnctic dipole of magnctic moment Pin =1A${\overrightarrow{P}}_{\text{in }}=1\overrightarrow{A}$ where, 1 is the current and A$\overrightarrow{A}$ is the area of the loop.Magnetic SusceptibilityIn simple isotropic substances the intensity of magnetization M$\overrightarrow{M}$ ) is found to be proportional to magnetizing field (H)$(\overrightarrow{\mathrm{H}})$ and parallel to it ie,a

M∝H (Magnitude)

M=𝜒nH

$$\begin{array}{l}\mathrm{M}\propto \mathrm{H}\text{ (Magnitude) }\\ \mathrm{M}={𝜒}_{\mathrm{n}}\mathrm{H}\end{array}$$Where 𝜒n${𝜒}_n$is dimensionless constant and is called the magnctio susceptibility. So, it can be defined as the ratio of intensity of magnetization (M) to the magnetizing field (H). It is simply the characteristic of the medium.Classification of Magnetic MaterialThere are three types of magnetic materials. They are:1. Diamagnetic material2. Paramagnetic material3. Ferromagnetic materialDiamagnetic materialThose substances which are feebly magnetized in the direction opposite to the applied field are called diamagnetic material. Examples of diamagnetic materials are bismuth, copper, water, mercury, alcohol, argon, gold,tin, mercury, antimony etc. The magnetic moment of atoms of a diamagnetic material is zero. They acquire induced dipole moments when the material placed in an external magnetic field. These moments are in opposite in the direction to the applied field.Some properties1. The diamagnetic materials are repelled by magnets.2. When a diamagnetic liquid in a watch glass is placed over two closely spaced pole pieces of the magnet, it is depressed at the middle while in the case of pole pieces separated by a distance, it rises at the middle. Similarly, when a diamagnetic liquid is placed in a U-tube and one of the limbs of the tube is placed between the two strong pole pieces of magnet, the liquid depressed at that limb.3. The diamagnetic materials move from a stronger to a weaker field.4. A diamagnetic rod, freely suspended in a magnetic field, slowly turns to set at right angle to the applied field.5. Since magnetized is opposite in direction to an applied field, the diamagnetic materials have the small value for the intensity of magnetization, I.6. The materials have always negative magnetic susceptibility, and accounts from -10-6 to -10-5.7. These materials are independent of temperature.Paramagnetic MaterialThose materials which are weekly magnetized in the same direction of the applied magnetic field are called paramagnetic material. The examples of paramagnetic materials are aluminum, chromium, oxygen, manganese, alkali, alkaline earth metal etc.The paramagnetic materials have permanent magnetic moments. These moments interacts weekly with each other and randomly orient in the different direction.Some Properties1. The paramagnetic materials are feebly attracted by magnets.2. A paramagnetic rod, freely suspended in a magnetic field, aligns along the field.3. The paramagnetic materials are temperature dependent and follow curve law.4. The relative permeability is nearly unity than ranges from 1.00001 to 1.003 for common ferromagnetic materials at room temperature. So, the magnetic lines of force inside the material placed in a magnetic field are more than that outside it.5. The susceptibility of paramagnetic substances has small positive value.Ferromagnetic MaterialThe ferromagnetic materials are highly magnetized in a magnetic field. The examples of ferromagnetic materials are iron, nickel and cobalt, and their alloys such as alnico. Gadolinium and dysprosium are ferromagnetic at low temperature.Some Properties1. Ferromagnetic materials are highly attracted by magnets.2. Ferromagnetic materials more from weaker to stronger field.3. A ferromagnetic rod, freely suspended in a magnetic field, turns fast to set along the applied field.4. The magnetic susceptibility is positive and very high and varies with applied field.5. The relative permeability is very high in the order of 1000 to 100,000.6. Ferromagnetic dust in a watch glass, placed over two closely spaced pole-pieces of the magnet, increases at the middle, while pole piece is separated by a distance, depresses in the middle.HysteresisLagging of magnetic field B behind the magnetic field H in a ferromagnetic material taken through a cycle of magnetization is called the hysteresis.

Hysteresis loop in Ferromagnetic Material:

A ferromagnetic material (iron) is taken and is placed on an external field B0. Now the magnetic field B in the material is studied with the magnetic intensity H ( = B0

𝜇0${\displaystyle H(=\frac{B_0}{{𝜇}_0}}$) is compared. When H is increased the value of magnetic field B increases and reaches maximum value Bs at point A. The value of magnetic field doesnot increases on increase in H. Now lowering the value of H, decreases the magnetic field but doesnot become zero when H is made zero. Value of magnetic field remains BR. (Through AR). H is reversed, then magnetic field continues to decrease and becomes zero at point C. Further increasing the value of H, magnetic field changes its direction and reaches maximum in oppoiste direction at point D. Now again H is decreased and made zero at point E, again reversed the reaches 'F' and ultimately 'A'. The closed loop denotes: a) loss of energy during magnetization b) how strongly is the material magnetizedArea of the hysteresis loop is directly proportional to the elergy loss.The material with broad hysteresis loop with higher retentivity (OR) and coercivity (OC) is suitable for making permanent magnet. Greater area of loop means greater energy has been lost during magnetization thus greater work should be done to demagnetize it. (As in steel)- Suitable for making permanent magnet Less area of loop with low value of coercivity means magnetization can be destroyed easily. ( as in soft iron) - Suitable for making core of transformer.Material with greater value of HC are called magnetically "Hard".

(a) Hysteresis loop in steel (b) Hysteresis loop in soft iron:

Langevin Theory of DiamagnetismGive Langevin theory of diamagnetism and show that the susceptibility of diamagnetic substances is independent of temperature. According to theory of Langevin, in diamagnetic substances, the electrons associated with different atoms rotate in orbits oriented so. that the net magnetic dipole moment is zero. When a magnetic field is applied on the atom the net magnetic dipole moment is not zero, but is negative for these substances. Hence the magnetic susceptibility of diamagnetic substances is negative and independent of temperature.To show the value of susceptibility of diamagnetic substance is independent of temperature we consider an atom in which an electron is moving in a definite orbit of radius r$\mathrm{r}$. When the electron is moving in the fixed orbit, the centripetal force is balanced by the electrostatic force of attraction. The centripetal force of an electron moving in the circular orbit =mv2

r=mr𝜔2$=\frac{\mathrm{m}{\mathrm{v}}^2}{\mathrm{r}}=\mathrm{m}\mathrm{r}{𝜔}^2$[a

where m=9.1×10-31kg is the

mass of an electron, 𝜔

angular velocity r is the radius

]$\left[\begin{array}{l}\text{ where }\mathrm{m}=9.1\times {10}^{-31}\mathrm{k}\mathrm{g}\text{ is the }\\ \text{ mass of an electron, }𝜔\\ \text{ angular velocity }\mathrm{r}\text{ is the radius }\end{array}\right]$When the external magnetic field is applied perpendicular to the circular loop then additional Lorentz force exists and electron thove with the higher speed. If 𝜔0${𝜔}_0$ be the angular velocity after applying magnetic field then.mri𝜔2=mr𝜔20 + Bev${\displaystyle m{r}_i{𝜔}^2=mr{𝜔}_0^2+Bev}$Where B$B$ is value of magnetic field, e$e$ is charge on electron and v${\displaystyle v}$ is the velocity of electron.or, mri𝜔2-mri𝜔02=Bev$\mathrm{m}{\mathrm{r}}_{\mathrm{i}}{𝜔}^2-{\mathrm{m}\mathrm{r}}_i{{𝜔}_0}^2=Bev$ 4. mri(𝜔2-𝜔20)=Bev$\mathrm{m}{\mathrm{r}}_i({𝜔}^2-{𝜔}_0^2)=Bev$or, mri(𝜔-𝜔0)(𝜔+𝜔0)=Bev${\mathrm{m}\mathrm{r}}_i(𝜔-{𝜔}_0)(𝜔+{𝜔}_0)=Bev$If 𝜔≃𝜔0$𝜔\simeq {𝜔}_0$ then, mriΔ𝜔 2𝜔=Bev${\displaystyle m{r}_i\Delta 𝜔2𝜔=Bev}$or, 𝛥𝜔=Bev

mri2𝜔$𝛥𝜔=\frac{Bev}{m{r}_{i}2𝜔}$ or, 𝛥𝜔=Be

2m𝜔(v

ri)$𝛥𝜔=\frac{Be}{2m𝜔}\left(\frac{v}{r_i}\right)$ or, 𝛥𝜔=Be

2m𝜔𝜔=Be

2m$𝛥𝜔=\frac{Be}{2m𝜔}𝜔=\frac{Be}{2m}$ This is angular frequency, called Larmor Frequency. The electric current due to orbital motion of electron is I=-e

T;T=$\mathrm{I}=\frac{-\mathrm{e}}{\mathrm{T}};\mathrm{T}=$ time periodor, I=-e

2𝜋

𝜔=-e𝜔

2𝜋$\mathrm{I}=\frac{\mathbf{-}\mathrm{e}}{\frac{2\mathbf{𝜋}}{\mathbf{𝜔}}}=\frac{-\mathrm{e}𝜔}{2𝜋}$The magnetic dipole moment m$\overrightarrow{\mathrm{m}}$ due to this current is given by, m$\overrightarrow{\mathrm{m}}$ m=IA$\overrightarrow{\mathrm{m}}=\mathrm{I}\mathrm{A}$or, m=-e𝜔

2𝜋⋅𝜋r2i$\overrightarrow{\mathrm{m}}=\frac{-e\overrightarrow{𝜔}}{2𝜋}\cdot 𝜋{\mathrm{r}}_{\mathrm{i}}^2$or, m=e𝜔r21

2$\overrightarrow{m}=\frac{e\overrightarrow{𝜔}{\mathrm{r}}_1^2}{2}$The change in magnetic moment =𝛥m=-er2i𝛥𝜔

2$=𝛥\mathrm{m}=\frac{-e{\mathrm{r}}_i^2𝛥𝜔}{2}$∴𝛥m=-er2i𝛥𝜔

2$\therefore 𝛥m=\frac{-\mathrm{e}{\mathrm{r}}_i^2𝛥𝜔}{2}$From equation (i) and (iii), we get𝛥m=-er2i

2,Be

2m$𝛥m=\frac{-e{r}_i^2}{2},\frac{Be}{2m}$ or, 𝛥m=-e2r2iB

4m$𝛥m=\frac{-e^2{r}_i^{2}B}{4m}$N$\mathrm{N}$ is total number of atoms in the material then magnetization M=N ∑i𝛥mi$M=N\sum _i^{}𝛥m_i$or, M=-Ne2

4m∑r2iB=-Ne2

4m ∑ir2i(𝜇0H)[∵B=𝜇0H]$M=\frac{-N{e}^2}{4m}\sum r_i^{2}B=\frac{-N{e}^2}{4m}\sum _i^{}{r}_i^2({𝜇}_{0}H)[\because B={𝜇}_{0}H]$or, M

H=-𝜇0Ne2

4m ∑ir2i$\frac{M}{H}=\frac{-{𝜇}_{0}N{e}^2}{4m}\sum _i^{}{r}_i^2$∴x=-𝜇0Ne2

4m ∑irt$\therefore x=\frac{-{𝜇}_{0}N{e}^2}{4m}\sum _i^{}{r}_t$which shows that the magnetic susceptibility of diamagnetic substance is negative and is independent of temperature. Energy Dissipated in Ferromagnetic MaterialExpression for the energy dissipated per unit volume of the ferromagnetic material during each cycle of magnetization. When a ferromagnetic substance is subjected to an external magnetic field each atom with magnetic moment 'm' experience a torque which tends to rotate the atom in the field direction. To rotatethe atomic magnet some amount of work should be done and this amount of work is dissipated in the form of heat enerry known as hysteresis loss, The moment of energy dissipated per cycle of magnetization is given by the area of the hysteresis loop as shown in figure.Hysteresis loop:

When a ferromagnetic substance is subjected into the extemal magnetic field H$\overrightarrow{H}$ then all the dipoles orient along the field direction and achieve the saturation magnetization or a particularvalue of external magnetic field H$\overrightarrow{H}$. The saturation magnetization is denoted by A$A$ in the graph, above this extemal field magnetization remains constant. Now if we reduce the applied magnetic field H$\overrightarrow{H}$ toward the minimum value then there exists the magnetization even if H=0$\overrightarrow{\mathrm{H}}=0$ which is given by curve OB$\mathrm{O}\mathrm{B}$. This OB$\mathrm{O}\mathrm{B}$ is known as retentivity. Again if we further reduce the magnetic field intensity H$\overrightarrow{\mathrm{H}}$ towards the negative value to reduce B$\overrightarrow{B}$ to zero, then magnetic intensity H$\overrightarrow{H}$ at this point is OC$OC$ =$=$ coercivity. Now if H$\overrightarrow{\mathrm{H}}$ is further reduced upto the saturation value, the magnetization is reached at point D as shown in figure. And if it is further increased towards positive value then it describes a curve DEFA. This closed loop is known as hysteresis loop.From the figure,a

OB= Retentivity

OC= Coercivity

$$\begin{array}{l}\mathrm{O}\mathrm{B}=\text{ Retentivity }\\ \mathrm{O}\mathrm{C}=\text{ Coercivity }\end{array}$$The torque due to external field H$\overrightarrow{H}$ is𝜏=M×H$$\overrightarrow{𝜏}=\overrightarrow{\mathrm{M}}\times \overrightarrow{\mathrm{H}}$$where 𝜃$𝜃$ is the angle between M$\overrightarrow{M}$ and H$\overrightarrow{H}$. Energy gained by dipole is given byU=-M⋅H$$U=-\overrightarrow{M}\cdot \overrightarrow{H}$$Workdone by magnetic dipole moment is written as, dW=𝜏(-d𝜃) [a

Negative sign means work is

done against the field

]$\mathrm{d}\mathrm{W}=𝜏(-\mathrm{d}𝜃)\left[\begin{array}{l}\text{ Negative sign means work is }\\ \text{ done against the field }\end{array}\right]$ or. dW=MHsin𝜃(-d𝜃)$dW=MH\sin 𝜃(-d𝜃)$The total magnetization M can be written as,M=Nmcos𝜃[a

where cos𝜃 gives the

component along the field direction

]$$\mathbf{M}=\mathbf{N}\mathbf{m}\cos 𝜃\left[\begin{array}{l}\text{ where }\cos 𝜃\text{ gives the }\\ \mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{o}\mathrm{n}\mathrm{e}\mathrm{n}\mathrm{t}\text{ along the field direction }\end{array}\right]$$or, dM=Nm(-sin𝜃)d𝜃$dM=Nm(-\sin 𝜃)d𝜃$dM=Nmsin𝜃(-d𝜃)$$\mathrm{d}\mathrm{M}=\mathrm{N}\mathrm{m}\sin 𝜃(-\mathrm{d}𝜃)$$If there exist N$\mathbf{N}$ number of molecules per unit volume, then total amount of workdone is given by,a

dW=NmHsin𝜃(-d𝜃)

dW=H(Nmsin𝜃(-d𝜃))

$$\begin{array}{l}\mathrm{d}\mathrm{W}=\mathrm{N}\mathrm{m}\mathrm{H}\sin 𝜃(-\mathrm{d}𝜃)\\ \mathrm{d}\mathrm{W}=\mathrm{H}(\mathrm{N}\mathrm{m}\sin 𝜃(-\mathrm{d}𝜃))\end{array}$$From equation (ii) and (iii), we getdW=HdM$$\mathrm{d}\mathrm{W}=\mathrm{H}\mathrm{d}\mathrm{M}$$Total amount of workdone during one complete cycle is given by the closed integration.W=∮cdW=∮cHdM$$\mathrm{W}=\underset{c}{\overset{}{\oint }}\mathrm{d}\mathrm{W}=\underset{c}{\overset{}{\oint }}\mathrm{H}\mathrm{d}\mathrm{M}$$For ferromagnetic substance,B =𝜇0(H+M)${\displaystyle B={𝜇}_0(H+M)}$a

[a Where 𝜇0= permeability H= magnetic field M= Intensity of magnetization B= Megnetic flux density ]

$$\begin{array}{l}\\ \left[\begin{array}{c}\text{ Where }{𝜇}_0=\text{ permeability }\\ \mathrm{H}=\text{ magnetic field }\\ \mathrm{M}=\text{ Intensity of magnetization }\\ \mathrm{B}=\text{ Megnetic flux density }\end{array}\right]\end{array}$$Differentiating above equation, we getdB=𝜇0(dH+dM)$\mathrm{d}\mathrm{B}={𝜇}_0(\mathrm{d}\mathrm{H}+\mathrm{d}\mathrm{M})$ or, dB

𝜇0=dH+dM$\frac{\mathrm{d}\mathrm{B}}{{𝜇}_0}=\mathrm{d}\mathrm{H}+\mathrm{d}\mathrm{M}$ or, dM=dB

𝜇0-dH$\mathrm{d}\mathrm{M}=\frac{\mathrm{d}\mathrm{B}}{{𝜇}_0}-\mathrm{d}\mathrm{H}$or, dM=dB

𝜇b-dH$dM=\frac{dB}{{𝜇}_b}-dH$Substituzing this relation in equation (V), we get.a

W	=∮H(dB 𝜇0-dH)
	=∮HdB 𝜇0-HdH
W	=∮HdB 𝜇0[∵∮HdH=0, the area of H with width dH]
	=1 𝜇0∮HdB
∴W	=1 𝜇0∮HdB

$$\begin{array}{rl}\mathrm{W}& =\oint \mathrm{H}(\frac{\mathrm{d}\mathrm{B}}{{𝜇}_0}-\mathrm{d}\mathrm{H})\\ & =\oint \frac{\mathrm{H}\mathrm{d}\mathrm{B}}{{𝜇}_0}-\mathrm{H}\mathrm{d}\mathrm{H}\\ & \\ \mathrm{W}& =\oint \frac{\mathrm{H}\mathrm{d}\mathrm{B}}{{𝜇}_0}[\because \oint \mathrm{H}\mathrm{d}\mathrm{H}=0,\text{ the area of }\mathrm{H}\text{ with width }d\mathrm{H}]\\ & =\frac{1}{{𝜇}_0}\oint \mathrm{H}\mathrm{d}\mathrm{B}\\ \therefore \mathrm{W}& =\frac{1}{{𝜇}_0}\oint \mathrm{H}\mathrm{d}\mathrm{B}\end{array}$$This is the required expression of the energy dissipated per unit volume of the ferromagnetic material during each cycle of magnetization. Paramagnetic SusceptibilityIn the absence of external magnetic field, each dipole molecules of paramagnetic materials are randomly oriented and hence the total magnetic moment is equal to zero. If external magnetic field is subjected to each molecule dipole experiences torque t=$t=$ n×H$\overrightarrow{\mathrm{n}}\times \overrightarrow{\mathrm{H}}$, where m$\overrightarrow{\mathrm{m}}$ is the magnetic moment of the molecular dipole and H$\overrightarrow{\mathrm{H}}$ is external magnetic field. This torque tends to align the dipole along the field direction. If the applied magnetic: field intensity is strong enough then all the dipoles perfecily align along the field direction and they are said to achieve saturation magnetism. But if the supplied magnetic field intensity is moderate value then they inclined with some angle 𝜃$𝜃$ and is known as moderate value of magnetism. This is due to the thermal agitation. The number of molecular dipole along the field direction is given by Maxwell-Boltzmann law. The law states that the number of molecules is directly proportional to the e -U

KBT${\displaystyle e\frac{-U}{K_{B}T}}$ where KB$K_B$ is Boltzmann constant, T$T$ is temperature and U$U$ is the energy given. We know, the magnetic potential energy is given y$y$U=-m⋅H=-mHcos𝜃$$\mathrm{U}=-\overrightarrow{\mathrm{m}}\cdot \overrightarrow{\mathrm{H}}=-\mathrm{m}\mathrm{H}\cos 𝜃$$where 𝜃$𝜃$ is the angle between m$\overrightarrow{\mathrm{m}}$ and H$\overrightarrow{\mathrm{H}}$. Total molecular dipole within the solid angle Ω$\Omega$ and Ω+dΩ$\Omega +\mathrm{d}\Omega$ can be obtained as,H=𝜋∫0Ce-1

𝜆LTdΩ [where C is proportionality constant] $$H=\underset{0}{\overset{𝜋}{\int }}\mathrm{C}{\mathrm{e}}^{-\frac{1}{{𝜆}_{\mathrm{L}}T}}\mathrm{d}\Omega \text{ [where C is proportionality constant] }$$If m$\mathrm{m}$ be the molecular dipole then the component of this dipole along the field direction is given by mcos𝜃$\mathrm{m}\cos 𝜃$.∴$\therefore$ Total magnetic dipole moments due to all molecular magnets within the solid angle Ω$\Omega$ and Ω+dΩ$\Omega +\mathrm{d}\Omega$ isM=N𝜋∫0Ce-U

KBTm cos𝜃 dΩ

𝜋∫0Ce-U

KbTdΩ$\mathrm{M}=\frac{\mathrm{N}\underset{0}{\overset{𝜋}{\int }}\mathrm{C}{\mathrm{e}}^{-\frac{\mathrm{U}}{\mathrm{K}\mathrm{B}\mathrm{T}}}mcos𝜃d\Omega }{\underset{0}{\overset{𝜋}{\int }}\mathrm{C}{\mathrm{e}}^{-\frac{\mathrm{U}}{{\mathrm{K}}_{\mathrm{b}}\mathrm{T}}}d\Omega }$Let mH

KHT=x$\frac{mH}{K_{H}T}=x$ and cos𝜃=y⇒-sin𝜃d𝜃=dy$\cos 𝜃=y\Rightarrow -\sin 𝜃d𝜃=dy$If 𝜃=0,y=1$𝜃=0,y=1$, and if 𝜃=𝜋$𝜃=𝜋$, then y=-1$y=-1$Putting all these values in equation (i), we getM=Nm-1∫0e-xyy(-dy)

1∫0exy(-dy)=NmL(x)$$\mathrm{M}=\frac{\mathrm{N}\mathrm{m}\underset{0}{\overset{-1}{\int }}{\mathrm{e}}^{-xy}\mathrm{y}(-\mathrm{d}\mathrm{y})}{\underset{0}{\overset{1}{\int }}{\mathrm{e}}^{xy}(-\mathrm{d}\mathrm{y})}=\mathrm{N}\mathrm{m}\mathrm{L}(\mathrm{x})$$where L(x)=$L(x)=$ Coth x-1

x$x-\frac{1}{x}$ is the Langevin Debye functionThe value of L(x)=Cothx-1

x=1

x+x

3+…+-1

x$L(x)=\mathrm{C}\mathrm{o}\mathrm{t}\mathrm{h}x-\frac{1}{x}=\frac{1}{x}+\frac{x}{3}+\dots +\frac{-1}{x}$[On neglecting higher powers of x$x$ as x$x$ is very small]We get,∴ L(x)=x

3$\therefore L(x)=\frac{x}{3}$Now,M=Nmx

3=Nm(mH

3KBT)=Nm2H

3KBT[∵x=mH

KBT]$$\mathrm{M}=\mathrm{N}\mathrm{m}\frac{\mathrm{x}}{3}=\mathrm{N}\mathrm{m}\left(\frac{\mathrm{m}\mathrm{H}}{3{\mathrm{K}}_{\mathrm{B}}\mathrm{T}}\right)=\frac{\mathrm{N}{\mathrm{m}}^2\mathrm{H}}{3{\mathrm{K}}_{\mathrm{B}}\mathrm{T}}[\because \mathrm{x}=\frac{\mathrm{m}\mathrm{H}}{{\mathrm{K}}_{\mathrm{B}}\mathrm{T}}]$$or, M

H=Nm2

3KBT$\frac{\mathrm{M}}{\mathrm{H}}=\frac{\mathrm{N}{\mathrm{m}}^2}{3{\mathrm{K}}_{\mathrm{B}}\mathrm{T}}$or, 𝜒=Nm2

3KBT [∵𝜒=M

H]$𝜒=\frac{\mathrm{N}{\mathrm{m}}^2}{3{\mathrm{K}}_{\mathrm{B}}\mathrm{T}}[\because 𝜒=\frac{\mathrm{M}}{\mathrm{H}}]$∴ 𝜒para =Nm2

3KBT$\therefore {𝜒}_{\text{para }}=\frac{\mathrm{N}{\mathrm{m}}^2}{3{\mathrm{K}}_{\mathrm{B}}\mathrm{T}}$which shows that the magnetic susceptibility is inversely proportional to the absolute temperature.