Magnetic Materials (Diamagnetic, Paramagnetic and Ferromagnetic) Questions in English

(C) The magnetic moment $(M)$ of a ferromagnetic material decreases as the temperature $(T)$ increases due to the increased thermal agitation of the atomic dipoles.
As the temperature approaches the Curie temperature $(T_c)$,the magnetic moment decreases rapidly and becomes zero at $T = T_c$,where the material transitions from ferromagnetic to paramagnetic.
The curve that correctly represents this behavior is a non-linear decrease that reaches zero at the Curie temperature,which corresponds to option $(C)$.

(A) For a paramagnetic substance,the magnetic susceptibility $(\chi)$ is inversely proportional to the absolute temperature $(T)$ according to Curie's Law,given by $\chi = C/T$. It is independent of the magnetising field $(H)$ for moderate fields and temperatures. Therefore,the graph of $\chi$ versus $H$ is a straight line parallel to the $H$-axis,indicating that $\chi$ remains constant as $H$ varies. This corresponds to the graph shown in option $A$.

(A) For a ferromagnetic material,the magnetic susceptibility $(\chi)$ is related to the absolute temperature $(T)$ by the Curie-Weiss law for temperatures above the Curie temperature $(T_c)$:
$\chi = \frac{C}{T - T_c}$
where $C$ is the Curie constant.
As the temperature $T$ increases towards $T_c$,the susceptibility $\chi$ decreases rapidly. Below the Curie temperature,the material is ferromagnetic and exhibits high susceptibility. Above the Curie temperature,the material transitions into a paramagnetic state,and its susceptibility follows the inverse relationship with $(T - T_c)$. The correct graphical representation for this behavior is a curve that starts at a high value and decreases as $T$ increases,as shown in option $(A)$.

(C) For a ferromagnetic substance,the relative permeability $(\mu_r)$ is very large at low temperatures.
As the temperature $(T)$ increases,the thermal agitation of the atoms increases,which opposes the alignment of magnetic dipoles.
When the temperature reaches the Curie temperature $(T_c)$,the substance undergoes a phase transition from ferromagnetic to paramagnetic.
Above the Curie temperature $(T_c)$,the substance behaves as a paramagnetic material,for which the relative permeability $(\mu_r)$ is slightly greater than $1$ and remains nearly constant with further increases in temperature.
Curve $C$ correctly represents this behavior,where $\mu_r$ is high at low temperatures and decreases to a value slightly above $1$ at $T = T_c$,remaining constant thereafter.

(B) For a diamagnetic substance,the magnetic susceptibility $(\chi)$ is small and negative.
The relationship between the intensity of magnetisation $(I)$ and the magnetising field $(H)$ is given by $I = \chi H$.
Since $\chi$ is negative,$I$ is also negative for a positive $H$.
This implies that the graph of $I$ versus $H$ must lie in the fourth quadrant (where $I$ is negative and $H$ is positive).
Among the given options,the line $OC$ represents a linear relationship with a negative slope,which corresponds to the behavior of a diamagnetic substance.

(A) For a paramagnetic substance, the magnetization $M$ is directly proportional to the magnetising field $H$ at low fields and temperatures, following Curie's Law: $M = \chi H$, where $\chi$ is the magnetic susceptibility.
Since $\chi$ is a small positive constant for paramagnetic substances, the graph of $M$ versus $H$ is a straight line passing through the origin with a positive slope.
Among the given options, curve $A$ represents a linear relationship with a positive slope, which is characteristic of paramagnetic materials.
Therefore, the correct option is $A$.

(A) Saturation magnetization $(M_s)$ is defined as the net magnetic moment per unit volume when all atomic dipoles are aligned in the direction of the external magnetic field.
It is given by the formula: $M_s = n \cdot \mu$
Where:
$n = 2 \times 10^{26} \, m^{-3}$ (number density of dipoles)
$\mu = 1.5 \times 10^{-23} \, A \cdot m^2$ (magnetic moment of each dipole)
Substituting the values:
$M_s = (2 \times 10^{26}) \times (1.5 \times 10^{-23})$
$M_s = 3 \times 10^3 \, A/m$
Therefore,the saturation magnetization is $3 \times 10^3 \, A/m$.

(B) The Curie temperature $(T_C)$ is a characteristic property of ferromagnetic materials.
Below the Curie temperature,the material exhibits ferromagnetic properties due to the alignment of magnetic domains.
As the temperature increases and exceeds $T_C$,thermal agitation disrupts the alignment of these magnetic domains.
Consequently,the ferromagnetic material loses its spontaneous magnetization and transitions into a paramagnetic state.
Therefore,above the Curie temperature,a ferromagnetic material becomes a paramagnetic material.

(B) diamagnetic substance is characterized by its tendency to move from a region of stronger magnetic field to a region of weaker magnetic field.
Because the magnetic field is strongest near the poles of a bar magnet,a diamagnetic substance experiences a repulsive force from both the north pole and the south pole.
Therefore,it is repelled by both poles.

(C) Electromagnets are designed to be easily magnetized and demagnetized.
Soft iron is chosen for the core of electromagnets because it possesses low retentivity and low coercivity.
Low retentivity ensures that the material does not retain significant magnetism when the current is switched off.
Low coercivity ensures that the material can be easily demagnetized by a small reverse magnetic field.
Thus,soft iron is an ideal soft magnetic material for this purpose.

(C) Diamagnetic materials are those in which the individual atoms do not possess any permanent magnetic dipole moment.
When placed in an external magnetic field,they develop a weak induced magnetic moment in a direction opposite to the applied field.
Therefore,the net magnetic moment of a diamagnetic atom in the absence of an external magnetic field is $0$.

(A) Diamagnetic materials are feebly repelled by a magnet.
Paramagnetic materials are feebly attracted by a magnet.
Ferromagnetic materials are strongly attracted by a magnet.
Non-magnetic materials remain unaffected by a magnet.
Based on the observations:
$(i)$ $A$ is feebly repelled,so $A$ is diamagnetic.
$(ii)$ $B$ is feebly attracted,so $B$ is paramagnetic.
$(iii)$ $C$ is strongly attracted,so $C$ is ferromagnetic.
$(iv)$ $D$ remains unaffected,so $D$ is non-magnetic.
Therefore,the statement '$B$ is of a paramagnetic material' is true.

(D) Magnetic susceptibility,denoted by $\chi$,is a measure of how much a material will become magnetized in an applied magnetic field.
For diamagnetic materials,the magnetic susceptibility $\chi$ is always negative,which means they are weakly repelled by a magnetic field.
For paramagnetic materials,$\chi$ is small and positive.
For ferromagnetic materials,$\chi$ is large and positive.
Therefore,the magnetic susceptibility is negative only for diamagnetic materials.

(A) Magnetic retentivity determines the amount of magnetism remaining in a material after the external magnetizing field is removed. For electromagnets,it is essential that the magnetism vanishes quickly when the current is switched off,which requires low retentivity.
Magnetic permeability (related to high susceptibility) determines how easily a material can be magnetized by an external field. High permeability ensures that the material becomes strongly magnetized even with a weak magnetizing field.
Therefore,soft iron is preferred for electromagnets because it possesses high permeability and low retentivity.

(A) When the current in the electromagnet is switched on,it creates a non-uniform magnetic field. Diamagnetic materials are weakly repelled by magnetic fields and tend to move from regions of stronger magnetic field to regions of weaker magnetic field. As the rod is pushed upward against gravity,it gains gravitational potential energy. This energy is supplied by the external current source that maintains the magnetic field in the electromagnet.

(A) Ferromagnetic substances are strongly attracted by a magnetic field because they possess permanent magnetic domains that align easily with the external field.
Paramagnetic substances are weakly attracted by a magnetic field because they possess permanent magnetic dipoles that align weakly with the external field.
Diamagnetic substances are weakly repelled by a magnetic field because they develop an induced magnetic moment in a direction opposite to the applied field due to the orbital motion of electrons.
Therefore,the magnet will attract $N_1$ strongly,$N_2$ weakly,and repel $N_3$ weakly.

(C) For a diamagnetic material,the relative permeability $\mu_r$ is always less than $1$ (i.e.,$\mu_r < 1$).
For any dielectric material,the relative permittivity $\varepsilon_r$ is always greater than $1$ (i.e.,$\varepsilon_r > 1$).
Comparing these conditions with the given options,we find that option $C$ satisfies both conditions: $\varepsilon_r = 1.5 > 1$ and $\mu_r = 0.5 < 1$.

(A) The magnetic field intensity $H$ inside a long solenoid is given by $H = nI$,where $n$ is the number of turns per unit length and $I$ is the current. Since $n$ and $I$ do not change,the $H$ field remains nearly unchanged.
The magnetic induction $B$ is given by $B = \mu H = \mu_0 \mu_r H$. Here,$\mu_r$ is the relative permeability of the core material.
When the soft iron core is heated beyond the Curie temperature $(T_C)$,it undergoes a phase transition from a ferromagnetic state to a paramagnetic state.
For a ferromagnetic material,$\mu_r$ is very large (e.g.,$1000$). For a paramagnetic material,$\mu_r$ is very close to $1$.
Since $\mu_r$ drops drastically from a large value to approximately $1$,the magnetic field $B$ inside the solenoid decreases drastically.

(B) diamagnetic substance is characterized by a negative magnetic susceptibility $(\chi < 0)$.
When placed in an external magnetic field $(B_0)$, the substance develops an induced magnetic moment in the direction opposite to the applied field.
This results in the internal magnetic field $(B)$ being weaker than the external field $(B_0)$.
Mathematically, $B = \mu_0(H + M)$, where $M$ is the magnetization. Since $M$ is negative for diamagnetic materials, the net magnetic induction inside the substance decreases compared to the external field.

(A) The total dipole moment of the sample is given by $M = N \mu$,where $N$ is the number of dipoles and $\mu$ is the dipole moment of each atom. The maximum possible dipole moment is $M_{max} = (20 \times 10^{24}) \times (1.5 \times 10^{-23} \ J \ T^{-1}) = 300 \ J \ T^{-1}$.
According to Curie's law,the magnetization $M$ is proportional to $B/T$,i.e.,$M = C(B/T)$.
For the first case: $M_1 = 0.15 \times M_{max} = 0.15 \times 300 = 45 \ J \ T^{-1}$.
Using $M_1 = C(B_1/T_1)$,we get $45 = C(0.84 / 4.2) = C(0.2)$,so $C = 45 / 0.2 = 225 \ J \ K \ T^{-2}$.
For the second case: $M_2 = C(B_2/T_2) = 225 \times (0.98 / 2.8) = 225 \times 0.35 = 78.75 \ J \ T^{-1}$.
Rounding to the nearest value,we get $79 \ J \ T^{-1}$ (Note: The option $7.9$ seems to be a typo for $79$). Given the options,$7.9$ is the closest scaled value.

(C) When a paramagnetic substance is placed in a magnetic field,it develops a magnetic moment in the direction of the field,effectively becoming a magnet with poles that reinforce the external field.
Conversely,a diamagnetic substance develops a magnetic moment opposite to the direction of the field,effectively becoming a magnet with poles that oppose the external field.
When placed between the pole pieces of a strong magnet,the paramagnetic cylinder concentrates the magnetic field lines within itself,while the diamagnetic cylinder expels them.
Because the paramagnetic material acts as a source of magnetic flux and the diamagnetic material acts as a sink (or repeller) of magnetic flux,their induced poles at the adjacent faces will be opposite in nature.
Specifically,the face of the paramagnetic cylinder near the diamagnetic one will develop a pole that attracts the induced pole of the diamagnetic cylinder.
Therefore,they will attract each other regardless of whether they are placed with a common axis or side by side,as the induced magnetic moments always result in an attractive force between the two distinct types of materials in this configuration.

(D) For a ferromagnetic material, the magnetic permeability $\mu$ is not constant; it depends on the magnetising field intensity $(H)$.
Initially, as $H$ increases, the magnetic susceptibility $\chi_m$ and permeability $\mu$ increase rapidly.
As the material approaches magnetic saturation, the rate of increase of magnetization decreases, and $\mu$ reaches a maximum value.
Beyond this point, as $H$ increases further towards saturation, the permeability $\mu$ decreases and eventually approaches a constant value (the permeability of free space $\mu_0$).
Graph $D$ correctly represents this behavior, showing an initial increase in $\mu$ followed by a decrease as the material approaches saturation.
Therefore, the correct option is $D$.

(A) According to Curie's Law,the magnetization $M$ of a paramagnetic material is directly proportional to the applied magnetic field $H$ and inversely proportional to the absolute temperature $T$.
Mathematically,$M = C \cdot \frac{H}{T}$,where $C$ is the Curie constant.
From this relation,we can see that for a fixed magnetic field $H$,the magnetization $M$ is inversely proportional to the temperature $T$ $(M \propto \frac{1}{T})$.
This means that at a lower temperature,the material will show higher magnetization for the same applied field.
Looking at the graph,for a constant value of $H$,the magnetization $M$ is highest for $T_1$ and lowest for $T_3$.
Therefore,$T_1 < T_2 < T_3$.

(B) For a diamagnetic substance,the magnetic susceptibility $(\chi)$ is small and negative.
It arises due to the orbital motion of electrons,which is essentially independent of the thermal agitation of the atoms.
Therefore,the magnetic susceptibility $(\chi)$ of a diamagnetic substance does not change with temperature.
This is represented by a horizontal line parallel to the temperature axis in the negative region of the $\chi$ axis.

(B) permanent magnet must retain its magnetism even in the presence of external demagnetizing fields.
Retentivity is the ability of a material to retain magnetism after the external field is removed,which should be high.
Coercivity is the measure of the resistance of a ferromagnetic material to becoming demagnetized.
For a permanent magnet,the material must be magnetically hard,meaning it should have high retentivity and high coercivity to prevent easy demagnetization.
Therefore,high coercivity is a fundamental requirement.

(B) The intensity of magnetisation is given by the relation $I = \chi H$,where $\chi$ is the magnetic susceptibility of the material.
For diamagnetic substances,the magnetic susceptibility $\chi$ is negative and its magnitude is very small (i.e.,$|\chi| \ll 1$).
Since $I = \chi H$ and $\chi < 0$,the graph of $I$ versus $H$ will be a straight line passing through the origin with a negative slope.
Because the magnitude of $\chi$ is very small,the slope of the line must be small,meaning the line should be close to the $H$-axis in the fourth quadrant.
Among the given options,line $OC$ represents a small negative slope,which correctly describes the variation of $I$ with $H$ for a diamagnetic substance.
Therefore,option $B$ is the correct answer.

(C) The relation $H \propto I$ implies a linear relationship between the intensity of magnetization $(I)$ and the magnetic field intensity $(H)$.
For diamagnetic and paramagnetic substances,the relationship is linear,and the magnetic susceptibility $\chi = \frac{I}{H}$ is a constant.
However,for ferromagnetic substances,the relationship between $I$ and $H$ is non-linear and exhibits the phenomenon of hysteresis.
Therefore,the simple linear relation $\chi = \frac{I}{H}$ is not applicable to ferromagnetic substances.

(B) $1$. Ferromagnetic materials $(F)$ strongly attract magnetic field lines,causing them to crowd inside the material. Bar $A$ shows this behavior.
$2$. Diamagnetic materials $(D)$ repel magnetic field lines,causing them to bend away from the material. Bar $B$ shows this behavior.
$3$. Paramagnetic materials $(P)$ weakly attract magnetic field lines,causing them to slightly crowd inside the material. Bar $C$ shows this behavior.
Therefore,the correct correspondence is $A \leftrightarrow F, B \leftrightarrow D, C \leftrightarrow P$.

(B) superconductor exhibits the Meissner effect,which is the expulsion of magnetic field lines from its interior when it is cooled below its critical temperature in an external magnetic field.
This phenomenon makes a superconductor a perfect diamagnet.
For a perfect diamagnet,the magnetic susceptibility $\chi = -1$.
The magnetic field inside the material is given by $B_s = \mu_0(H + M)$. Since $M = \chi H = -H$,we get $B_s = \mu_0(H - H) = 0$.
Therefore,the magnetic field inside the superconductor is $B_s = 0$.

(A) According to Curie's Law,the magnetic susceptibility $\chi$ of a paramagnetic material is inversely proportional to its absolute temperature $T$,i.e.,$\chi \propto \frac{1}{T}$.
Therefore,we can write the relation as $\chi_1 T_1 = \chi_2 T_2$.
Given:
$\chi_1 = 2.8 \times 10^{-4}$
$T_1 = 350 \text{ K}$
$T_2 = 300 \text{ K}$
Substituting the values:
$\chi_2 = \frac{\chi_1 T_1}{T_2} = \frac{2.8 \times 10^{-4} \times 350}{300}$
$\chi_2 = \frac{2.8 \times 10^{-4} \times 7}{6}$
$\chi_2 = 3.2666... \times 10^{-4} \approx 3.267 \times 10^{-4}$.

(C) The magnetic moment $(M)$ of a ferromagnetic material decreases as the temperature $(T)$ increases. This decrease is non-linear,and the magnetic moment drops to zero at the Curie temperature $(T_C)$,where the material transitions from ferromagnetic to paramagnetic. The curve shown in option $C$ correctly depicts this non-linear behavior where $M$ gradually decreases and reaches zero at the Curie temperature.

(B) When a bar of soft iron is placed in a uniform magnetic field such that it is parallel to the field lines,the magnetic lines of force tend to concentrate inside the soft iron bar. This happens because soft iron is a ferromagnetic material with high magnetic permeability. Consequently,the magnetic field lines prefer to pass through the material rather than the surrounding air. This results in an increased density of magnetic field lines within the bar,as shown in Figure $B$.

(A) For a ferromagnetic material,the magnetic susceptibility $\left( \chi \right)$ is very large and positive. As the temperature $T$ increases,the thermal agitation of the atoms increases,which opposes the alignment of magnetic dipoles. Below the Curie temperature,the material remains ferromagnetic,but as it approaches the Curie temperature,the susceptibility decreases significantly. The variation is such that it remains relatively high and then drops as the temperature increases towards the Curie point,as shown in the provided figure.

(B) superconductor exhibits the Meissner effect,which means it expels magnetic fields from its interior.
For a perfect diamagnetic material,the internal magnetic field $B = 0$.
Since $B = \mu_0(H + M) = 0$,we have $M = -H$.
The magnetic susceptibility $\chi$ is defined as $M/H$.
Therefore,$\chi = -H/H = -1$.

(C) material used for making a permanent magnet should have high retentivity so that it remains strongly magnetized even after the external magnetic field is removed.
It should also have high coercivity so that it is not easily demagnetized by stray magnetic fields,temperature fluctuations,or minor mechanical impacts.
Steel has higher coercivity compared to soft iron,which makes it more suitable for permanent magnets.
Soft iron,on the other hand,has low coercivity and is therefore preferred for electromagnets where rapid magnetization and demagnetization are required.

(B) According to the Curie-Weiss law for ferromagnetic materials above the Curie temperature,the magnetic susceptibility $\chi_m$ is inversely proportional to the temperature difference $(T - T_c)$. However,in the context of standard textbook problems where Curie's law $\chi_m \propto \frac{1}{T}$ is applied to magnetic materials:
Given:
$T_1 = 30\,^{\circ}\text{C} = 30 + 273 = 303\,\text{K}$
$T_2 = 333\,^{\circ}\text{C} = 333 + 273 = 606\,\text{K}$
Using the relation $\chi_m \propto \frac{1}{T}$:
$\frac{\chi_1}{\chi_2} = \frac{T_2}{T_1}$
Substituting the values:
$\frac{\chi}{\chi_2} = \frac{606}{303} = 2$
Therefore,$\chi_2 = \frac{\chi}{2} = 0.5\,\chi$.

(C) The Curie temperature $(T_C)$ is the temperature above which a ferromagnetic material loses its spontaneous magnetization and behaves as a paramagnetic material.
As the temperature increases beyond the Curie point,the thermal agitation disrupts the alignment of magnetic dipoles,causing the material to transition from a ferromagnetic state to a paramagnetic state.

(B) Ferromagnetic materials are characterized by strong attraction to magnets,which means they tend to move from a region of weak magnetic field to a region of strong magnetic field.
Option $B$ states that they move from a strong to a weak magnetic field,which is a characteristic of diamagnetic materials,not ferromagnetic materials.
Therefore,option $B$ is the incorrect characteristic for ferromagnetic materials.

(B) When a liquid is placed in a non-uniform magnetic field,it experiences a force depending on its magnetic susceptibility.
Paramagnetic liquids are attracted towards the stronger part of the magnetic field.
Since the meniscus of the liquid rises in the limb when placed between the poles of an electromagnet (where the field is stronger),it indicates that the liquid is attracted to the magnetic field.
Therefore,the liquid is paramagnetic.

(D) For diamagnetic materials:
$1$. The relative permeability $\mu_r$ is slightly less than $1$ $(\mu_r < 1)$.
$2$. The magnetic susceptibility $\chi$ is small and negative $(-1 \le \chi < 0)$.
$3$. The magnetic susceptibility $\chi$ is independent of temperature,meaning it does not change with variations in temperature.
Since all three statements are correct,the correct option is $D$.

(A) For a magnetic material,the relative permeability $\mu_r$ is related to the magnetic susceptibility $\chi_m$ by the equation $\mu_r = 1 + \chi_m$.
An ideal diamagnetic substance is a perfect diamagnet,which exhibits the Meissner effect.
In a perfect diamagnet,the magnetic field inside the material is zero,meaning $B = 0$.
Since $B = \mu_0 H(1 + \chi_m)$,for $B$ to be $0$,we must have $1 + \chi_m = 0$.
Therefore,the magnetic susceptibility of an ideal diamagnetic substance is $\chi_m = -1$.

(A) Superconductors exhibit the Meissner effect,which means they expel magnetic fields from their interior when cooled below their critical temperature $(T_c)$.
Because of this,they act as perfect diamagnets.
When a magnet is placed near a superconductor,the superconductor generates surface currents that create a magnetic field opposing the magnet's field.
This results in a repulsive force that allows the magnet to levitate above the superconductor.
Therefore,the Assertion is correct because the magnet levitates,and the Reason is correct because superconductors exhibit a repulsive force towards magnets due to the Meissner effect.

(C) Diamagnetic materials exhibit a weak magnetism in the direction opposite to the applied magnetic field.
Therefore,the Assertion is correct.
However,diamagnetic materials do not have a permanent magnetic dipole moment because all electrons are paired,resulting in a net magnetic moment of zero.
Therefore,the Reason is incorrect.
Thus,the correct option is $C$.

(A) The magnetic susceptibility of ferromagnetic materials decreases as temperature increases.
At a specific transition temperature known as the Curie temperature $(T_C)$,ferromagnetic materials transition into paramagnetic materials.
This happens because,at high temperatures,the thermal agitation (kinetic energy) of the atoms becomes strong enough to overcome the exchange coupling forces that align the magnetic moments within the domains.
As a result,the ordered structure of the magnetic domains is destroyed,leading to the loss of ferromagnetism.
Therefore,both the Assertion and the Reason are correct,and the Reason provides a valid explanation for the Assertion.

(A) The magnetic susceptibility of a ferromagnetic substance does not follow a simple linear relationship with temperature like paramagnetic substances.
Instead,it decreases in a complex manner as temperature increases.
Curie's law states that susceptibility $\chi \propto 1/T$.
Ferromagnetic substances only begin to obey this law after they are heated above their Curie temperature $(T_C)$,where they transition into a paramagnetic state.
Therefore,the assertion that they do not obey Curie's law (in their ferromagnetic state) is correct,and the reason explains the transition at the Curie point.

(C) According to Curie's Law,the magnetic susceptibility of a paramagnetic material is inversely proportional to its absolute temperature,given by $\chi = C/T$.
As the temperature $T$ decreases,the susceptibility $\chi$ increases,leading to greater magnetisation for the same external magnetic field.
This occurs because at lower temperatures,the random thermal motion that disrupts the alignment of atomic dipoles is reduced,making it easier for the external magnetic field to align them.
Therefore,the Assertion is correct,but the Reason is incorrect because magnetisation in paramagnetic materials is strongly dependent on temperature.

(A) Electromagnets are magnets that can be turned on and off by controlling the electric current.
For an electromagnet,the core material undergoes cyclic magnetization and demagnetization. To minimize energy loss during these cycles,the material should have a small hysteresis loop area,which implies low coercivity and low retentivity.
Soft iron is characterized by high magnetic permeability,high susceptibility,and low coercivity. Because of its low coercivity,it can be easily magnetized and demagnetized,making it the ideal material for electromagnets.
Therefore,both the Assertion and the Reason are correct,and the Reason is the correct explanation for the Assertion.

(D) Number of atomic dipoles,$n = 2.0 \times 10^{24}$.
Dipole moment of each atomic dipole,$M = 1.5 \times 10^{-23} \; J \, T^{-1}$.
Total dipole moment of the sample if fully saturated,$M_{\text{total}} = n \times M = 2.0 \times 10^{24} \times 1.5 \times 10^{-23} = 30 \; J \, T^{-1}$.
Given saturation at $B_1 = 0.64 \; T$ and $T_1 = 4.2 \; K$ is $15 \%$.
Effective dipole moment $M_1 = 0.15 \times 30 = 4.5 \; J \, T^{-1}$.
According to Curie's law,the magnetization $M \propto \frac{B}{T}$,so $\frac{M_2}{M_1} = \frac{B_2}{B_1} \times \frac{T_1}{T_2}$.
Substituting the values for $B_2 = 0.98 \; T$ and $T_2 = 2.8 \; K$:
$M_2 = M_1 \times \frac{B_2}{B_1} \times \frac{T_1}{T_2} = 4.5 \times \frac{0.98}{0.64} \times \frac{4.2}{2.8}$.
$M_2 = 4.5 \times 1.53125 \times 1.5 = 10.336 \; J \, T^{-1}$.

(N/A) Magnetic susceptibility $\chi$ is defined as the ratio of the intensity of magnetization $M$ to the magnetic intensity $H$, i.e., $\chi = M/H$.
$(i)$ For paramagnetic materials, the atoms have a permanent magnetic dipole moment. When placed in an external magnetic field, they get weakly magnetized in the direction of the field. Thus, the magnetic susceptibility $\chi$ is small and positive ($0 < \chi < \epsilon$, where $\epsilon$ is a small positive number).
$(ii)$ For diamagnetic materials, the atoms develop an induced magnetic dipole moment in a direction opposite to the applied magnetic field. Thus, they are weakly repelled by the field. The magnetic susceptibility $\chi$ is small and negative ($-1 \le \chi < 0$).