Magnetic Classification of Substances

Nickel

Reason — Nickel is strongly magnetised in the direction of the magnetising field and is strongly attracted by a magnet. Hence, it is a ferromagnetic substance.

Neon

Reason — In a neon atom all the electrons are paired with opposite spins, so the net magnetic dipole moment of the atom is zero. In a magnetic field it is feebly magnetised opposite to the field, so neon is diamagnetic.

Copper

Reason — Copper is diamagnetic. A diamagnetic substance is feebly magnetised in a direction opposite to the magnetising field, so it is repelled by an external magnetic field.

fall

Reason — In a non-uniform magnetic field a diamagnetic substance tends to move from the stronger to the weaker part of the field. The field is strongest between the poles, so the solution is pushed out of that region and its level in that arm is depressed.

Fig. B

Reason — Soft iron is ferromagnetic and is strongly magnetised in the direction of the field. Inside the bar its own field adds to the original field, so the lines of force are drawn into the bar and become concentrated within it, as shown in Fig. B.

it becomes paramagnetic

Reason — Above the Curie temperature, thermal agitation destroys the alignment within the domains. The ferromagnetic property suddenly disappears and the substance behaves as a paramagnetic substance.

X is diamagnetic and Y is paramagnetic

Reason — The relative permeability μ_r is slightly less than 1 for diamagnetic substances and slightly more than 1 for paramagnetic substances, whereas it is very large for ferromagnetic substances.

Assertion is false, and Reason is true.

Reason — Due to the strong mutual (exchange) interaction between neighbouring atoms, the atomic magnets are aligned in the same direction only within a domain. Different domains are randomly oriented, so in the material as a whole all the atomic magnets do not have their dipole moments in the same direction.

Water

Reason — A relative permeability between 0 and 1 (slightly less than 1) is a property of diamagnetic substances. Water is diamagnetic.

Copper

Reason — Permeability less than μ₀ means μ_r < 1, which is the case for diamagnetic substances. Copper is diamagnetic.

The atoms of a paramagnetic material have a net non-zero (permanent) magnetic dipole moment of their own. In the atoms of a diamagnetic material, the electrons form pairs with opposite spins, so the net magnetic dipole moment of each atom is zero.

Suspend each rod freely in a magnetic field. The paramagnetic rod sets itself with its axis parallel to the field, whereas the diamagnetic rod sets itself perpendicular to the field.

It increases with increasing applied magnetic field and decreases with increasing temperature.

Explanation:
By Curie's law, $\text M = \text C\frac{\text H}{\text T}$, so the magnetisation is directly proportional to the magnetic intensity H and inversely proportional to the kelvin temperature T, as long as H/T does not become too large.

Given,

• Magnetic susceptibility, χ_m = 0.0001

The relative permeability is related to the magnetic susceptibility by

$$\begin{aligned}\mu_\text r = 1 + \chi_\text m \\[1em] = 1 + 0.0001 \\[1em] \mu_\text r = 1.0001\end{aligned}$$

Hence, the relative permeability of the material is 1.0001.

The area of the hysteresis loop of soft iron is smaller than that of steel, so the hysteresis loss in soft iron is smaller. As the core of a transformer goes through complete cycles of magnetisation continuously, a soft-iron core wastes less energy and is heated less.

The iron in the earth's core is in the molten state, at a temperature far above its Curie temperature (770°C). Above the Curie temperature iron becomes paramagnetic and cannot retain magnetism, so it cannot be the source of the earth's magnetism.

The region is surrounded by a soft-iron ring (or enclosure). Due to its high permeability, the lines of force of any external magnetic field are drawn into the soft iron and pass through it, leaving the enclosed region free from the field.

Magnetisation (M) : The magnetisation (intensity of magnetisation) of a magnetised substance is the magnetic moment per unit volume of the substance,

$\vec{\text M} = \frac{\vec{\text m}}{\text V}$

where $\vec{\text m} \space$is the magnetic moment of the substance of volume V. Its SI unit is A m^-1.

Magnetic susceptibility (χ_m) : It is the ratio of the intensity of magnetisation M to the magnetic intensity H of the magnetising field,

$\chi_\text m = \frac{\text M}{\text H}$

It is a measure of how easily a substance is magnetised in a magnetising field, and is a dimensionless number.

Iron and nickel.

Explanation:
Iron and nickel are strongly magnetised in the direction of the magnetising field. Of the others, copper, silver and gold are diamagnetic, while platinum and aluminium are paramagnetic.

The magnetic permeability of a substance is the ratio of the magnetic induction B inside the magnetised substance to the magnetic intensity H of the magnetising field,

$\mu = \frac{\text B}{\text H}$

It measures the conduction of magnetic lines of force through the substance. Its SI unit is N A^-2 (or T m A^-1).

The relative permeability of a substance is the ratio of the magnetic permeability μ of the substance to the permeability of free space μ₀,

$\mu_\text r = \frac{\mu}{\mu_0}$

It is a dimensionless quantity. It is related to the magnetic susceptibility by

$\mu_\text r = 1 + \chi_\text m$

As the susceptibility is small and negative, the material is diamagnetic.

In a uniform magnetic field the specimen is feebly magnetised opposite to the field, so the lines of force are pushed out and become less dense inside it than outside, as shown.

In a non-uniform magnetic field, the specimen tends to move from the stronger to the weaker part of the field.

A paramagnetic material is feebly magnetised in the direction of the field, so the lines of force crowd into it and become slightly more dense inside it than outside, as shown.

Each atom of iron has a net magnetic moment and behaves like a tiny magnet. Due to strong mutual interaction, these atomic magnets form domains, each magnetised to saturation. Normally the domains are randomly oriented, so their external effects cancel.

When the rod is placed along the axis of the solenoid, its magnetic field acts on the domains. The domains favourably oriented with respect to the field grow in size at the expense of others, and the remaining domains rotate until their magnetisation lies along the field. Hence, the rod is magnetised along the axis of the solenoid.

Yes. The atoms (or ions) of a paramagnetic salt are tiny magnetic dipoles. In a magnetic field they tend to align with the field, but thermal agitation disturbs this alignment. Higher the temperature, greater the agitation and smaller the magnetisation; on cooling, the alignment improves and the magnetisation increases. By Curie's law, $\text M = \text C\frac{\text H}{\text T}$.

In a diamagnetic substance the electrons form pairs whose magnetic moments cancel, so its atoms have no permanent magnetic moment to be aligned. In a magnetising field, one electron of each pair is accelerated and the other slowed down, inducing a small moment opposite to the field. This induced moment does not depend on the alignment of atomic magnets, so thermal agitation, and hence temperature, has no effect on it.

No.

Explanation:
The Curie temperature of iron is 770°C. At 1000°C the violent thermal agitation destroys the domain alignment and the iron becomes paramagnetic, losing its magnetism. On cooling in a field-free space, the domains form again but are randomly oriented, as no external field is present to align them. So the iron is not magnetised.

Since χ_m = M/H, for a given magnetising field the intensity of magnetisation of iron is much greater than that of aluminium. Thus, iron is magnetised much more easily than aluminium: iron is ferromagnetic, while aluminium is only paramagnetic.

The magnetisation of a ferromagnetic material is not completely reversible. In a magnetising field it is magnetised by the motion of the domain boundaries and the rotation of the domains. When the field is reduced to zero, the domains do not return completely to their original state, so some residual magnetisation remains (hysteresis). The material thus keeps a "memory" of its magnetisation, which is used for the magnetic storage of information, as on magnetic tapes and computer discs.

A high coercivity means that a large reverse magnetising field is needed to destroy the residual magnetism. Hence, the magnetisation of a permanent magnet is not wiped out by stray external fields, mechanical ill-treatment or temperature changes.

The substance should have:

(i) high retentivity, so that the magnet is strong;

(ii) high coercivity, so that its magnetisation is not easily wiped out;

(iii) high permeability.

Example : Steel (cobalt steel), or alloys such as alnico (Al + Ni + Co).

On the basis of their behaviour in a magnetic field, materials are classified as :

(a) Diamagnetic — feebly magnetised opposite to the magnetising field and feebly repelled by a magnet (e.g. bismuth, copper).

(b) Paramagnetic — feebly magnetised in the direction of the field and feebly attracted by a magnet (e.g. aluminium, platinum).

(c) Ferromagnetic — strongly magnetised in the direction of the field and strongly attracted by a magnet (e.g. iron, nickel).

Iron is a ferromagnetic substance.

(i) As the temperature rises, the ferromagnetism of iron decreases. At its Curie temperature, 770°C, the ferromagnetic property suddenly disappears and iron becomes paramagnetic.

(ii) The atoms of a paramagnetic salt are tiny magnetic dipoles which tend to align along the applied field, while thermal agitation disturbs this alignment. On cooling, the thermal agitation decreases, so more dipoles align with the field and the magnetisation increases, in accordance with Curie's law, $\text M = \text C\frac{\text H}{\text T}$.

The magnetism of a paramagnetic substance is due to the alignment of its atomic magnets along the magnetising field. On heating, the thermal agitation of the atoms increases and disturbs this alignment, so the magnetisation decreases (M ∝ 1/T). Hence, its magnetism is lost on heating.

Atomic model of magnetism : Every atom has a positively charged nucleus around which electrons revolve in discrete orbits. Each revolving electron is equivalent to a tiny current loop and gives a magnetic dipole moment to the atom. Each electron also spins about its own axis, and this spin too produces a magnetic dipole moment. Most of the magnetic moment of an atom is due to electron spin; the contribution of orbital revolution is very small.

Diamagnetic substances : The electrons form pairs with opposite spins, so the net magnetic moment of each atom is zero. In an external field $\vec{\text B}$, one electron of each pair is slowed down and the other accelerated, so a small moment is induced opposite to $\vec{\text B}$. The substance is feebly magnetised opposite to the field.

Paramagnetic substances : The atoms have an excess of electrons spinning in the same direction, so each atom has a net magnetic moment. Normally these atomic magnets are randomly oriented and the net moment is zero. In an external field each experiences a torque and tends to align along the field, but thermal agitation opposes this. The substance is feebly magnetised along the field.

Ferromagnetic substances : The atoms have net magnetic moments, and due to strong mutual interaction groups of atoms form domains in which all the atomic moments are aligned. Each domain is magnetically saturated. In an external field the domains grow and rotate into the field direction, so the substance is strongly magnetised along the field.

(i) Correct.

Since μ_r = 1 + χ, the susceptibility is

$\chi = \mu_\text r - 1$

(ii) Incorrect.

For a paramagnetic substance, χ ∝ 1/T (Curie's law). Decreasing the temperature increases the susceptibility. To decrease it, the temperature of the solution should be increased.

(iii) Correct.

A paramagnetic substance tends to move from the weaker to the stronger part of a magnetic field. The field is strongest between the poles, so the solution is drawn into that region and its level in that arm rises.

Curie's law : The magnetisation M of a paramagnetic substance is directly proportional to the magnetic intensity H of the magnetising field and inversely proportional to the kelvin temperature T,

$\text M = \text C\frac{\text H}{\text T} \qquad ...(\text i)$

where C is the Curie constant. Since $\chi_\text m = \frac{\text M}{\text H}$,

$\chi_\text m = \frac{\text C}{\text T}$

that is, the susceptibility of a paramagnetic substance varies inversely as its kelvin temperature. The law holds as long as H/T is not too large.

Variation of χ_m with temperature :

(a) Diamagnetic : χ_m is negative and does not change with temperature; the graph is a straight line parallel to the T-axis, below it.

(b) Paramagnetic : χ_m is positive and varies as 1/T; the graph is a rectangular hyperbola.

(c) Ferromagnetic : χ_m is very large and decreases with temperature in a complex way up to the Curie temperature T_C (ferromagnetic region). Above T_C the substance becomes paramagnetic and follows the Curie-Weiss law,

$\chi_\text m = \frac{\text C}{\text T - \text T_\text C}$

Intensity of magnetisation (M) : It is the magnetic moment per unit volume of a magnetised substance,

$\vec{\text M} = \frac{\vec{\text m}}{\text V}$

Its SI unit is A m^-1.

Magnetic intensity (H) : It expresses the capability of the magnetising field to magnetise a substance, and is defined by the vector relation

$\vec{\text H} = \frac{\vec{\text B}}{\mu_0} - \vec{\text M}$

where $\vec{\text B} \space$is the magnetic induction inside the substance and μ₀ is the permeability of free space. Its SI unit is A m^-1.

Relation χ = (μ_r − 1) :

By definition of the magnetic intensity H of the magnetising field,

$$\begin{aligned}\text H = \frac{\text B}{\mu_0} - \text M \\[1em] \text B = \mu_0(\text H + \text M) \qquad ...(\text i)\end{aligned}$$

For a substance of magnetic susceptibility χ,

$\text M = \chi\text H \qquad ...(\text{ii})$

Substituting Eq. (ii) in Eq. (i),

$\text B = \mu_0(1 + \chi)\text H \qquad ...(\text{iii})$

But B = μH, where μ is the permeability of the substance. Comparing with Eq. (iii),

$\mu = \mu_0(1 + \chi)$

Since $\frac{\mu}{\mu_0}$ is the relative permeability μ_r,

$$\begin{aligned}\mu_\text r = 1 + \chi \\[1em] \chi = \mu_\text r - 1\end{aligned}$$

1 oersted = 80 A m^-1 :

The oersted is the CGS unit of H. From the relation $\text H = \frac{\text B}{\mu_0} - \text M$, with 1 gauss = 10^-4 T and μ₀ = 4π × 10^-7 T m A^-1,

$$\begin{aligned}1\ \text{oersted} = \frac{1\ \text{gauss}}{\mu_0} \\[1em] = \frac{10^{-4}\ \text T}{4\pi \times 10^{-7}\ \text{T m A}^{-1}} \\[1em] = \frac{10^3}{4\pi}\ \text{A m}^{-1} \\[1em] = 79.6\ \text{A m}^{-1} \\[1em] 1\ \text{oersted} \approx 80\ \text{A m}^{-1}\end{aligned}$$

Hence, 1 oersted = 80 ampere/metre.

The figure shows the variation of the magnetisation M (or the flux density B) of a ferromagnetic substance with the magnetic intensity H. The point O represents the unmagnetised state.

(i) Magnetic saturation : As H is increased from zero, M increases non-uniformly along OA. At A the substance reaches a state in which any further increase in H produces no increase in M. This state is called magnetic saturation.

(ii) Residual magnetisation (retentivity) : When H is decreased, M decreases along a new path AB, not along AO. When H becomes zero, a magnetisation OB still remains; this is the residual magnetism. The retentivity of a substance is a measure of the magnetisation remaining in it when the magnetising field is removed.

(iii) Coercive force : When H is increased in the reverse direction, M decreases along BC and becomes zero at C. The reverse magnetising field OC needed to reduce the magnetisation to zero is the coercive force (coercivity). It is a measure of the reverse magnetising field required to destroy the residual magnetism.

(iv) Hysteresis : On increasing H further in the reverse direction, the substance saturates at D, and on taking H back to its positive maximum the curve DEFA is traced. Throughout the cycle, M (or B) lags behind H. This lagging of M (or B) behind H is called hysteresis, and the closed curve ABCDEFA is the hysteresis loop. The energy lost per unit volume of the substance in one complete cycle of magnetisation is equal to the area of the loop.

Magnetisation (M) : The magnetic moment per unit volume of a magnetised substance, $\vec{\text M} = \frac{\vec{\text m}}{\text V}$. It represents the extent to which the substance is magnetised; SI unit A m^-1.

Magnetising field intensity (H) : The capability of the magnetising field to magnetise a substance, defined by $\vec{\text H} = \frac{\vec{\text B}}{\mu_0} - \vec{\text M}$; SI unit A m^-1.

Permeability (μ) : The ratio of the magnetic induction B inside the magnetised substance to the magnetic intensity H of the magnetising field, $\mu = \frac{\text B}{\text H}$. It measures the conduction of magnetic lines of force through the substance; SI unit N A^-2.

Relative permeability (μ_r) : The ratio of the permeability μ of the substance to the permeability of free space μ₀, $\mu_\text r = \frac{\mu}{\mu_0}$. It is dimensionless and equal to 1 for vacuum.

Magnetic susceptibility (χ_m) : The ratio of the intensity of magnetisation to the magnetic intensity of the magnetising field, $\chi_\text m = \frac{\text M}{\text H}$. It is dimensionless and measures how easily a substance is magnetised.

Relation between permeability and susceptibility :

From the definition of H, B = μ₀(H + M). Putting M = χ_mH,

$\text B = \mu_0(1 + \chi_\text m)\text H$

Since B = μH,

$$\begin{aligned}\mu = \mu_0(1 + \chi_\text m) \\[1em] \mu_\text r = 1 + \chi_\text m\end{aligned}$$

Origin of domains : In a ferromagnetic substance, as in a paramagnetic one, each atom has a net non-zero magnetic moment and behaves like a tiny magnet. But due to a strong mutual interaction, the atoms form a large number of small effective regions called domains. Each domain contains about 10¹⁷ to 10²¹ atoms whose magnetic axes are aligned in the same direction. Thus, even without any external field, each domain is magnetically saturated and is a strong magnet.

Unmagnetised state : In the normal state, the different domains are randomly oriented [Fig. (a)], so their resultant magnetic moment in any direction is zero. This is why every piece of iron is not a magnet.

Magnetisation in an external field : When the substance is placed in an external magnetic field, its magnetic moment increases in two ways :

(i) Displacement of domain boundaries — the domains oriented favourably with respect to the field grow in size, while those oriented opposite to it shrink [Fig. (b)]. This is the main process in weak fields.

(ii) Rotation of domains — the domains rotate until their magnetic moments are aligned more or less along the field [Fig. (c)]. This is the main process in strong fields.

The field lines inside the substance therefore become much more dense than outside, and the substance is strongly magnetised.

Residual magnetism : When the external field is removed, the domain boundaries do not move completely back to their original positions, so the substance is not completely demagnetised and some residual magnetism remains (hysteresis).

Demagnetisation : If the magnet is heated above its Curie temperature or hammered, the thermal agitation of the atoms becomes so vigorous that the domains are totally disturbed. The resultant magnetic moment becomes almost zero and the substance behaves as a paramagnetic substance above the Curie temperature.

Given,

• Relative permeability of iron, μ_r = 5500

The relative permeability μ_r and the magnetic susceptibility χ_m of a substance are related as

$\mu_\text r = 1 + \chi_\text m$

Therefore,

$$\begin{aligned}\chi_\text m = \mu_\text r - 1 \\[1em] = 5500 - 1 \\[1em] \chi_\text m = 5499\end{aligned}$$

Hence, the magnetic susceptibility of iron is 5499.

Given,

• Length of the rod, l = 5.0 cm = 5.0 × 10^-2 m
• Diameter of the rod = 1.0 cm, so radius r = 0.5 cm = 0.5 × 10^-2 m
• Intensity of magnetisation, M = 5.3 × 10³ A m^-1

Volume of the cylindrical rod,

$$\begin{aligned}\text V = \pi \text r^2 \text l \\[1em] = 3.14 \times \left(0.5 \times 10^{-2}\right)^2 \times 5.0 \times 10^{-2} \\[1em] = 3.14 \times 0.25 \times 10^{-4} \times 5.0 \times 10^{-2} \\[1em] \text V = 3.925 \times 10^{-6}\ \text m^3\end{aligned}$$

The intensity of magnetisation is the magnetic moment per unit volume, $\text M = \frac{\text m}{\text V}$. Hence the magnetic moment is

$$\begin{aligned}\text m = \text M \times \text V \\[1em] = 5.3 \times 10^3 \times 3.925 \times 10^{-6} \\[1em] = 20.80 \times 10^{-3} \\[1em] \text m \approx 20.8 \times 10^{-3}\ \text{A m}^2\end{aligned}$$

Hence, the magnetic moment of the rod magnet is 20.8 × 10^-3 A m².

Given,

• Magnetic intensity of the magnetising field, H = 2000 A m^-1
• Magnetic field induction in the material, B = 1.0 N A^-1 m^-1
• μ₀ = 4π × 10^-7 N A^-2

(i) Magnetic permeability :

$$\begin{aligned}\mu = \frac{\text B}{\text H} \\[1em] = \frac{1.0}{2000} \\[1em] \mu = 5 \times 10^{-4}\ \text{N A}^{-2}\end{aligned}$$

(ii) Relative permeability :

$$\begin{aligned}\mu_\text r = \frac{\mu}{\mu_0} \\[1em] = \frac{5 \times 10^{-4}}{4 \times 3.14 \times 10^{-7}} \\[1em] = \frac{5 \times 10^{-4}}{12.56 \times 10^{-7}} \\[1em] = 398.1 \\[1em] \mu_\text r \approx 398\end{aligned}$$

(iii) Magnetic susceptibility :

$$\begin{aligned}\chi_\text m = \mu_\text r - 1 \\[1em] = 398 - 1 \\[1em] \chi_\text m = 397\end{aligned}$$

(iv) Intensity of magnetisation :

$$\begin{aligned}\text M = \chi_\text m \text H \\[1em] = 397 \times 2000 \\[1em] = 794000 \\[1em] \text M = 7.94 \times 10^5\ \text{A m}^{-1}\end{aligned}$$

Hence, for the material, (i) μ = 5 × 10^-4 N A^-2, (ii) μ_r = 398, (iii) χ_m = 397 and (iv) M = 7.94 × 10⁵ A m^-1.

Given,

• Magnetic moment, m₁ = 5.0 A m² at T₁ = 4 K in B₁ = 1.0 T
• T₂ = 3 K, B₂ = 1.5 T

By Curie's law, the magnetisation of a paramagnetic substance is directly proportional to the magnetising field and inversely proportional to the kelvin temperature. Hence the magnetic moment acquired by the salt is

$$\begin{aligned}\text m \propto \frac{\text B}{\text T} \\[1em] \frac{\text m_2}{\text m_1} = \frac{\text B_2}{\text B_1} \times \frac{\text T_1}{\text T_2}\end{aligned}$$

Therefore,

$$\begin{aligned}\text m_2 = \text m_1 \times \frac{\text B_2}{\text B_1} \times \frac{\text T_1}{\text T_2} \\[1em] = 5.0 \times \frac{1.5}{1.0} \times \frac{4}{3} \\[1em] = \frac{30}{3} \\[1em] \text m_2 = 10\ \text{A m}^2\end{aligned}$$

Hence, the magnetic moment of the salt at 3 K in a field of 1.5 T is 10 A m².

Given,

• χ₁ = 1.2 × 10⁵ at T₁ = 300 K
• χ₂ = 1.44 × 10⁵

Magnesium is paramagnetic. By Curie's law, its magnetic susceptibility varies inversely as the kelvin temperature,

$$\begin{aligned}\chi_\text m \propto \frac{1}{\text T} \\[1em] \chi_1 \text T_1 = \chi_2 \text T_2\end{aligned}$$

Therefore,

$$\begin{aligned}\text T_2 = \frac{\chi_1 \text T_1}{\chi_2} \\[1em] = \frac{1.2 \times 10^5 \times 300}{1.44 \times 10^5} \\[1em] = \frac{360}{1.44} \\[1em] \text T_2 = 250\ \text K\end{aligned}$$

Hence, the magnetic susceptibility of magnesium becomes 1.44 × 10⁵ at 250 K.

Note: The textbook prints both susceptibilities as × 10⁵; for a paramagnetic substance like magnesium they should read × 10^-5. The answer depends only on their ratio, so it is unaffected.

The figure shows the hysteresis loop ABCDEFA of a ferromagnetic substance, with the magnetising field (magnetic intensity) H along the x-axis and the intensity of magnetisation M along the y-axis. Along OA the substance is magnetised from the unmagnetised state and at A it reaches magnetic saturation.

• Retentivity : OB (or OE) is the magnetisation remaining in the substance when the magnetising field H is reduced to zero.
• Coercive force (coercivity) : OC (or OF) is the reverse magnetising field required to reduce the residual magnetisation to zero.

Area of the loop : The area of the hysteresis loop represents the energy lost (as heat) per unit volume of the substance in one complete cycle of magnetisation, called the hysteresis loss.

The magnetic susceptibility of a ferromagnetic material decreases when it is heated.

Explanation:
Ferromagnetism decreases with rise in temperature. At the Curie temperature T_c the ferromagnetic property disappears and the material becomes paramagnetic. Above T_c its susceptibility falls with temperature according to the Curie-Weiss law, $\chi = \frac{\text C}{\text T - \text T_\text c}$.

Ferromagnetic > paramagnetic > diamagnetic.

Explanation:
Ferromagnetic materials have a very large positive susceptibility, paramagnetic materials a small positive susceptibility and diamagnetic materials a small negative susceptibility.

(i) A paramagnetic substance is one which, when placed in a magnetising field, is feebly magnetised in the direction of the field, and so is feebly attracted towards a magnet. Examples : aluminium, platinum, sodium.

Curie's law : The magnetic susceptibility of a paramagnetic substance is inversely proportional to its kelvin temperature T,

$\chi = \frac{\text C}{\text T}$

where C is the Curie constant.

(ii) Given,

• Relative permeability of aluminium, μ_r = 1.000022

Using the relation μ_r = 1 + χ,

$$\begin{aligned}\chi = \mu_\text r - 1 \\[1em] = 1.000022 - 1 \\[1em] \chi = 0.000022\end{aligned}$$

Hence, the magnetic susceptibility of aluminium is 0.000022 (= 2.2 × 10^-5).

Curie temperature.

Explanation:
The temperature above which a ferromagnetic substance loses its ferromagnetic property and becomes paramagnetic is called its Curie temperature. For iron it is 770°C.

A is paramagnetic and B is diamagnetic.

Explanation:
The relative permeability μ_r is slightly more than 1 for paramagnetic substances and slightly less than 1 for diamagnetic substances. For ferromagnetic substances it is very large, of the order of hundreds and thousands.

• Diamagnetic materials : χ is small and negative.
• Paramagnetic materials : χ is small and positive.
• Ferromagnetic materials : χ is very large and positive.

(i) When a ferromagnetic substance is taken through a cycle of magnetisation, its intensity of magnetisation M (and also the magnetic flux density B) always lags behind the magnetising field H. This lagging of M (or B) behind H is called hysteresis.

(ii) The three elements of the earth's magnetic field are :

• angle of declination
• angle of dip (or inclination)
• horizontal component of the earth's magnetic field.

The Curie temperature of a ferromagnetic substance is the temperature above which the substance loses its ferromagnetic property and becomes paramagnetic. For example, the Curie temperature of iron is 770°C and that of nickel is 358°C.

1.0001

Reason — Using μ_r = 1 + χ with χ = 0.0001 gives μ_r = 1.0001. Platinum is paramagnetic, so its relative permeability is slightly more than 1.

Curie's law : The magnetisation M of a paramagnetic substance is directly proportional to the magnetic intensity H of the magnetising field and inversely proportional to its kelvin temperature T,

$\text M = \text C\frac{\text H}{\text T}$

where C is the Curie constant. Since χ = M/H, the law can also be written as

$\chi = \frac{\text C}{\text T} \quad \text{or} \quad \chi \propto \frac{1}{\text T}$

Physical basis : The atoms of a paramagnetic substance have a net magnetic moment of their own and behave like tiny bar-magnets. The external field tends to align them along its direction, while thermal agitation disturbs this alignment. On cooling, thermal agitation decreases, the alignment improves and the magnetisation (and hence χ) increases; on heating, the reverse happens.

The law holds only so long as H/T is not too large, because M cannot exceed the value corresponding to complete alignment of all the atomic magnets.

a ferromagnetic substance behaves like a paramagnetic substance.

Reason — Ferromagnetism decreases with rise in temperature. Above the Curie temperature the ferromagnetic property disappears and the substance becomes paramagnetic.

μ_r = 1 + χ

Explanation:
Inside a magnetised substance, B = μ₀(H + M). Putting M = χH,

$\text B = \mu_0(1 + \chi)\text H$

Since B = μH, μ = μ₀(1 + χ). Dividing by μ₀,

$$\begin{aligned}\frac{\mu}{\mu_0} = 1 + \chi \\[1em] \mu_\text r = 1 + \chi\end{aligned}$$

(i) The core of a transformer is taken through complete cycles of magnetisation continuously. Its material must therefore have a low hysteresis loss, so that the core is not heated much, and a high permeability, so that a large flux density is obtained at a low magnetising field. The hysteresis loop of soft iron is much narrower than that of steel, so its hysteresis loss per cycle is smaller, and its permeability is greater than that of steel. Hence, the core is made of soft iron and not of steel.

(ii) It means that soft iron is ferromagnetic only below 770°C. When it is heated above 770°C, its ferromagnetic property disappears and it behaves as a paramagnetic substance.

diamagnetic material only

Reason — A diamagnetic material is magnetised opposite to the magnetising field, so M and H are opposite and χ = M/H is negative. For paramagnetic and ferromagnetic materials χ is positive.

B for electromagnets and transformers

Reason — The cores of electromagnets and transformers go through repeated cycles of magnetisation, so they need a material with small hysteresis loss (loop of small area) and low retentivity and coercivity, so that they are easily magnetised and demagnetised. Loop B is narrower than loop A, so material B suits them.

the current source

Reason — On switching on the current, the energy of the magnetic field set up between the poles is supplied by the current source. The diamagnetic rod tends to move from the stronger to the weaker part of the field and is pushed up; the work done in raising it also comes from the current source.

3.3 × 10^-4

Reason — Volume of the cube, V = (10^-2 m)³ = 10^-6 m³. Using M = m/V and χ = M/H,

$$\begin{aligned}\text M = \frac{20 \times 10^{-6}}{10^{-6}} \\[1em] \text M = 20\ \text{A m}^{-1} \\[1em] \chi = \frac{20}{60 \times 10^3} \\[1em] \chi = 3.3 \times 10^{-4}\end{aligned}$$

2600 A/m

Reason — The magnet is demagnetised when the magnetic intensity H = nI = (N/l)I of the solenoid equals its coercivity.

$$\begin{aligned}\text H = \frac{100}{0.2} \times 5.2 \\[1em] \text H = 2600\ \text{A/m}\end{aligned}$$

$\sqrt{\frac{\mu_0}{\varepsilon_0}}$

Reason — With [μ₀] = [MLT^-2A^-2] and [ε₀] = [M^-1L^-3T⁴A²],

$$\begin{aligned}\left[\frac{\mu_0}{\varepsilon_0}\right] = \frac{[\text{MLT}^{-2}\text A^{-2}]}{[\text M^{-1}\text L^{-3}\text T^4\text A^2]} \\[1em] = [\text M^2\text L^4\text T^{-6}\text A^{-4}] \\[1em] \left[\sqrt{\frac{\mu_0}{\varepsilon_0}}\right] = [\text{ML}^2\text T^{-3}\text A^{-2}]\end{aligned}$$

This is the dimensional formula of resistance (R = V/I).

3.267 × 10^-4

Reason — By Curie's law, χ ∝ 1/T.

$$\begin{aligned}\frac{\chi_2}{\chi_1} = \frac{\text T_1}{\text T_2} \\[1em] \chi_2 = 2.8 \times 10^{-4} \times \frac{350}{300} \\[1em] \chi_2 = 3.267 \times 10^{-4}\end{aligned}$$

P : Large retentivity, large coercivity

Reason — A permanent magnet must be strong (high retentivity) and its magnetisation must not be wiped out by stray fields, mechanical ill-treatment or temperature changes (high coercivity). A transformer core instead needs low hysteresis loss and high permeability.

0.75 A/m

Reason — By Curie's law, the magnetisation is proportional to the applied field and inversely proportional to the kelvin temperature.

$$\begin{aligned}\frac{\text M_2}{\text M_1} = \frac{\text B_2}{\text B_1} \times \frac{\text T_1}{\text T_2} \\[1em] \text M_2 = 6 \times \frac{0.3}{0.4} \times \frac{4}{24} \\[1em] \text M_2 = 0.75\ \text{A/m}\end{aligned}$$

(A), (C)

Reason — For a diamagnetic material χ is small and negative, so M = χH is a straight line through the origin with negative slope — graph (A). Its susceptibility is independent of temperature, so χ versus T is a straight line parallel to the T-axis below it — graph (C).

Statement I is true but Statement II is false.

Reason — Ferromagnetism decreases with rise in temperature and above the Curie temperature a ferromagnet becomes paramagnetic, so Statement I is true. At high temperature thermal agitation disorients the magnetic dipoles and the domains are destroyed; the domain wall area does not increase, so Statement II is false.

(C) and (E)

Reason — The field lines of a toroid are completely confined within its core (C), and a perfect diamagnetic material has χ = −1 (E).

Note: Statement (B) is also correct, so option 3 likewise lists only correct statements; the answer given follows the printed key (d).

may increase or decrease in size and change its orientation

Reason — In an external field, the domains oriented favourably with respect to the field increase in size, those oriented opposite to it decrease in size, and the domains also rotate towards the direction of the field.

a macroscopic region with saturation magnetisation

Reason — Below the Curie temperature, the magnetic moments of the 10¹⁷ to 10²¹ atoms of a domain are aligned in the same direction by mutual interaction, so each domain is in a state of magnetic saturation even without an external field.

high permeability and low retentivity

Reason — The core of an electromagnet must give a strong field at a low magnetising current (high permeability) and must lose its magnetism when the current is switched off (low retentivity).

Both Statement I and Statement II are true.

Reason — For paramagnetic substances $\chi = \frac{\text C}{\text T}$, and for ferromagnetic substances above the Curie temperature $\chi = \frac{\text C}{\text T - \text T_\text c}$; in both, χ increases as the temperature decreases. In diamagnetic substances the external field modifies the motion of the paired electrons (Lenz's law), inducing magnetic moments opposite to the applied field. So both statements are true.

4π × 10^-5

Reason — Using μ = μ₀(1 + χ),

$$\begin{aligned}\mu = 4\pi \times 10^{-7} \times (1 + 99) \\[1em] \mu = 4\pi \times 10^{-5}\ \text{Wb/A-m}\end{aligned}$$

zero

Reason — Magnetic monopoles do not exist and magnetic field lines form closed loops, so every line entering a closed surface also leaves it. Hence, the net magnetic flux through any closed surface is zero (Gauss's law in magnetism).

Both A and R are correct and R is the correct explanation of A.

Reason — The core of an electromagnet must have high permeability, to give a strong field at low magnetising fields, and low retentivity, so that it loses its magnetism when the current is switched off. Soft iron has both properties, which is why electromagnets are made of it.

Both Statement I and Statement II are true.

Reason — A diamagnetic substance has a small negative susceptibility, whose lowest value is −1 for a perfect diamagnet (superconductor), so −1 ≤ χ < 0. In a non-uniform field, a diamagnetic substance tends to move from the stronger to the weaker part of the field.

A-II, B-III, C-IV, D-I

Reason — Diamagnetic materials have small negative χ with −1 as its lower limit (II); ferromagnetic materials have very large positive χ (III); paramagnetic materials have small positive χ (IV); a non-magnetic medium such as vacuum has χ = 0 (I).

Given,

• Number of turns per unit length, n = 1000 turns/m
• Initial relative permeability, μ_r1 = 500
• Final relative permeability, μ_r2 = 750
• Volume of core, V = 10³ cm³
• Current, I = 0.75 A

The magnetic intensity inside the solenoid, H = nI, is the same in both cases. The magnetic moment of the core is

$$\begin{aligned}\text m = \text{MV} \\[1em] = \chi\text{HV} \\[1em] \text m = (\mu_\text r - 1)\text{nIV}\end{aligned}$$

Since n, I and V do not change, m ∝ (μ_r − 1). Hence, the fractional change in the magnetic moment is

$$\begin{aligned}\frac{\Delta \text m}{\text m} = \frac{(\mu_\text{r2} - 1) - (\mu_\text{r1} - 1)}{\mu_\text{r1} - 1} \\[1em] = \frac{749 - 499}{499} \\[1em] \frac{\Delta \text m}{\text m} = \frac{250}{499}\end{aligned}$$

Comparing with $\frac{\text x}{499}$,

Hence, the value of x is 250.

Given,

• Susceptibility of aluminium, χ = 2.2 × 10^-5

The magnetic intensity H = nI inside the toroid is unchanged when aluminium is filled in. Without aluminium B₀ = μ₀H, and with aluminium B = μ₀(1 + χ)H, so

$$\begin{aligned}\text B = \text B_0(1 + \chi) \\[1em] \frac{\text B - \text B_0}{\text B_0} = \chi\end{aligned}$$

Percentage increase in the magnetic field,

$$\begin{aligned}\frac{\text B - \text B_0}{\text B_0} \times 100 = 2.2 \times 10^{-5} \times 100 \\[1em] = 2.2 \times 10^{-3} \\[1em] = \frac{22}{10^4}\end{aligned}$$

Hence, the value of x is 22.

Given,

• Coercivity, H = 5 × 10³ A/m
• Length of solenoid, l = 30 cm = 0.3 m
• Number of turns, N = 150

The magnet gets demagnetised when the magnetic intensity inside the solenoid equals its coercivity, H = $\frac{\text N}{\text l}$ I. Therefore,

$$\begin{aligned}\text I = \frac{\text H\text l}{\text N} \\[1em] = \frac{5 \times 10^3 \times 0.3}{150} \\[1em] \text I = 10\ \text A\end{aligned}$$

Hence, the current required is 10 A.

If assertion is true but reason is false.

Explanation

Assertion (A) is correct: In an atom, each electron revolving in its orbit acts as a tiny current loop and gives a magnetic dipole moment; each electron also spins about its own axis, which gives a further magnetic moment. The magnetic moment of the atom is the vector sum of both.

Reason (R) is incorrect: A charged particle at rest produces no magnetic field; only a moving charge produces a magnetic field.

Therefore, assertion is true but reason is false.

If both assertion and reason are false.

Explanation

Assertion (A) is incorrect: The temperature inside the earth's core is far above the Curie temperature of iron (770°C), so the iron there cannot behave as a magnet. The earth's magnetic field is attributed to electric currents circulating in the molten iron and other conducting materials of the core.

Reason (R) is incorrect: A magnet loses its magnetism at a high temperature (above its Curie temperature), not at a low temperature. On cooling, thermal agitation decreases and magnetisation improves.

Therefore, both assertion and reason are false.

If both assertion and reason are false.

Explanation

Assertion (A) is incorrect: A compass needle is free to rotate only in a horizontal plane. At the magnetic north pole the earth's field is vertical, so its horizontal component is zero and the needle experiences no aligning torque in the horizontal plane. It therefore rests in any direction and does not rotate in a vertical direction.

Reason (R) is incorrect: At the magnetic poles the earth's magnetic field has only a vertical component; the horizontal component is zero.

Therefore, both assertion and reason are false.

If both assertion and reason are true and reason is the correct explanation of assertion.

Explanation

Assertion (A) is correct: At a neutral point the net magnetic field is zero, so the compass needle experiences no torque and can stay pointing in any direction.

Reason (R) is also correct: At a neutral point the horizontal field of the magnet exactly balances the horizontal component of the earth's field, which is why the net field there is zero and the needle points in any arbitrary direction.

Therefore, both assertion and reason are true and reason is the correct explanation of assertion.

If both assertion and reason are true and reason is the correct explanation of assertion.

Explanation

Assertion (A) is correct: The torque on a bar magnet of magnetic moment m in a uniform field B is τ = mB sin θ. It is maximum (= mB) when θ = 90°, that is, when the field is perpendicular to the magnetic moment.

Reason (R) is also correct: τ = mB sin θ depends on the angle θ between the field and the magnetic dipole moment, and this dependence explains why the torque is maximum at θ = 90°.

Therefore, both assertion and reason are true and reason is the correct explanation of assertion.

If both assertion and reason are false.

Explanation

Assertion (A) is incorrect: Copper is a diamagnetic substance; it is feebly repelled by a magnet.

Reason (R) is incorrect: Paramagnetic substances are feebly magnetised in the direction of the magnetising field. It is diamagnetic substances that are magnetised opposite to the field.

Therefore, both assertion and reason are false.

If both assertion and reason are true but reason is not the correct explanation of assertion.

Explanation

Assertion (A) is correct: Below the Curie temperature, the susceptibility of a ferromagnetic substance is very large and changes with temperature in a very complex way, so it does not follow Curie's law χ = C/T. Above the Curie temperature it follows the Curie-Weiss law $\chi = \frac{\text C}{\text T - \text T_\text c}$.

Reason (R) is also correct: At the Curie temperature the ferromagnetic property disappears and the substance starts behaving as paramagnetic. This describes the change at the Curie point but does not explain why Curie's law fails for ferromagnetic substances.

Therefore, both assertion and reason are true but reason is not the correct explanation of assertion.

If both assertion and reason are true and reason is the correct explanation of assertion.

Explanation

Assertion (A) is correct: A transformer core is taken through cycles of magnetisation continuously, so its material must have a low hysteresis loss. Soft iron meets this requirement and is used for transformer cores.

Reason (R) is also correct: The energy lost per unit volume in each cycle equals the area of the hysteresis loop. Soft iron has a narrow loop, hence a small hysteresis loss, which explains its use as a transformer core.

Therefore, both assertion and reason are true and reason is the correct explanation of assertion.

If both assertion and reason are false.

Explanation

Assertion (A) is incorrect: Cobalt steel has high retentivity and high coercivity, so it is a suitable material for permanent magnets.

Reason (R) is incorrect: A permanent magnet should possess a high retentivity (so that it is strong) and a high coercivity (so that its magnetism is not easily wiped out).

Therefore, both assertion and reason are false.

If both assertion and reason are true and reason is the correct explanation of assertion.

Explanation

Assertion (A) is correct: The core of an electromagnet should have high permeability and low retentivity, which soft iron has; so electromagnets are made of soft iron.

Reason (R) is also correct: Because the coercivity of soft iron is small, the core is easily demagnetised and loses its magnetism when the current is switched off, which is what an electromagnet requires. This explains the assertion.

Therefore, both assertion and reason are true and reason is the correct explanation of assertion.

If both assertion and reason are true but reason is not the correct explanation of assertion.

Explanation

Assertion (A) is correct: For a perfectly diamagnetic substance χ = −1, so μ_r = 1 + χ = 0 and μ = μ_rμ₀ = 0.

Reason (R) is also correct: Permeability is indeed a measure of how readily a material conducts magnetic lines of force through it, but this definition does not show why its value is zero for a perfect diamagnet; that follows from χ = −1.

Therefore, both assertion and reason are true but reason is not the correct explanation of assertion.

If assertion is true but reason is false.

Explanation

Assertion (A) is correct: By Curie's law, the susceptibility of a paramagnetic substance is χ = C/T, so χT = C, a constant.

Reason (R) is incorrect: The susceptibility of a paramagnetic substance varies inversely as its kelvin temperature; it is the susceptibility of a diamagnetic substance that is independent of temperature.

Therefore, assertion is true but reason is false.

If both assertion and reason are true and reason is the correct explanation of assertion.

Explanation

Assertion (A) is correct: Above the Curie temperature the ferromagnetic property disappears and the substance becomes paramagnetic.

Reason (R) is also correct: At high temperature, thermal agitation becomes so vigorous that the alignment of atomic magnets within the domains is destroyed, so the substance is left with only individual atomic magnets, as in a paramagnetic substance. This explains the assertion.

Therefore, both assertion and reason are true and reason is the correct explanation of assertion.

If both assertion and reason are true and reason is the correct explanation of assertion.

Explanation

Assertion (A) is correct: When water is placed in a magnetic field, the flux density within it is slightly less than in free space.

Reason (R) is also correct: Water is diamagnetic, so it is feebly magnetised opposite to the field and its relative permeability is slightly less than 1. This explains the decrease of the field in water.

Therefore, both assertion and reason are true and reason is the correct explanation of assertion.

If assertion is true but reason is false.

Explanation

Assertion (A) is correct: Magnetic susceptibility χ = M/H, and M and H have the same unit (A m^-1), so χ is a pure number.

Reason (R) is incorrect: There can be no magnetisation in vacuum, so the magnetic susceptibility of vacuum is zero, not one.

Therefore, assertion is true but reason is false.

If both assertion and reason are true and reason is the correct explanation of assertion.

Explanation

Assertion (A) is correct: If a bar magnet is broken, each piece becomes a complete magnet with its own north and south poles; an isolated pole cannot be obtained.

Reason (R) is also correct: Magnetic monopoles do not exist; magnetic field lines always form closed loops. This is why the poles of a magnet cannot be separated.

Therefore, both assertion and reason are true and reason is the correct explanation of assertion.