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Electricity and Magnetism

Paramagnetism

Paramagnetic substances are weakly attracted by external magnetic fields and possess a small positive magnetic susceptibility. Their atomic magnetic moments possess permanent dipole moments that tend to align weakly with an applied magnetic field.

UP TGT ScienceUP PGT PhysicsStudy Notes + Practice

Scope and prerequisites

TGT core

Basic definitions of magnetic materials, susceptibility and permeability relations, qualitative behavior of paramagnetic substances with temperature.

PGT extension

Curie's Law, Curie-Weiss Law, quantum theory of paramagnetism, Langevin function, and molecular field theory.

Prerequisites

Magnetic induction, magnetic field intensity, and magnetization vector
Magnetic susceptibility and relative permeability
Thermal energy and Boltzmann distribution

Origin of Paramagnetism

Paramagnetic materials consist of atoms, ions, or molecules having a net permanent magnetic dipole moment due to partially filled inner electron shells (such as d or f orbitals).
In the absence of an external magnetic field, thermal agitation randomizes the orientations of these atomic dipoles, resulting in a net macroscopic magnetization of zero.
When an external magnetic field is applied, it exerts a torque on the atomic dipoles, tending to align them along the field direction against thermal randomization.

Magnetic Susceptibility and Permeability

Magnetic susceptibility (χ) for a paramagnetic substance is small and positive, typically ranging from 10^-5 to 10^-2.
The relative permeability (μ_r) is slightly greater than unity, given by the relation μ_r = 1 + χ.
Magnetic induction B inside a paramagnetic material is slightly larger than the applied vacuum magnetic field B₀, expressed as $B = μ_{0}(H + M) = μH$.

Curie's Law of Paramagnetism

Pierre Curie experimentally discovered that the magnetic susceptibility of a paramagnetic substance is inversely proportional to its absolute temperature T.
The mathematical expression is χ = C / T, where C is the material-specific Curie constant.
This law holds well at moderate temperatures and weak magnetic fields where thermal energy (k_BT) significantly exceeds the magnetic potential energy.

Curie-Weiss Law

At temperatures close to absolute zero or in the presence of strong molecular interactions, Curie's law deviates.
The Curie-Weiss law modifies the temperature dependence as χ = C / (T - θ), where θ is the Curie-Weiss temperature.
For pure paramagnets, θ is close to zero, whereas for ferromagnetic materials transitioning to paramagnetic behavior above the Curie point, θ equals the Curie temperature.

Derivations

Classical Langevin Theory of Paramagnetism

Conditions: Applied magnetic field B, absolute temperature T, N magnetic dipoles per unit volume each of magnetic moment μ_m.

Consider a magnetic dipole in a magnetic field B at an angle θ. The potential energy is $U = -μ_{m}B \cosθ$.
According to Maxwell-Boltzmann statistics, the probability of a dipole pointing in the solid angle dΩ is proportional to exp(-U/k_BT) dΩ = exp(μ_mB cosθ / k_BT) 2π sinθ dθ.
The magnetization M is given by $M = N μ_{m} <\cosθ>$, where the average value is evaluated using the partition function integration.
This yields Langevin's function L(x) = coth(x) - (1/x), where $x = μ_{m}B / k_{B}T$.
For weak fields and high temperatures (x << 1), the series expansion gives L(x) ≈ x/3, leading to $M = N μ_{m}^{2} B / (3 k_{B}T)$.

Result: Susceptibility χ = μ₀ N μ_m² / (3 k_BT) = C / T

Relation between Relative Permeability and Susceptibility

Conditions: Uniform magnetic field inside a linear isotropic paramagnetic medium.

Start with the fundamental magnetic field vector relation: $B = μ_{0}(H + M)$.
Substitute the definition of magnetic susceptibility, M = χ H, into the equation.
Factor out the magnetic field intensity H: $B = μ_{0}H(1 + χ)$.
Compare this with the macroscopic definition B = μH, where μ is the absolute permeability.

Result: μ_r = μ / μ₀ = 1 + χ

Formula sheet

Formula / law	Equation	Variables and units	Conditions
Magnetic Susceptibility Definition	$$\chi = \frac{M}{H}$$	M = Magnetization, H = Magnetic field intensity Dimensionless	Linear isotropic media
Relative Permeability	$$\mu_r = 1 + \chi$$	\m$u_r = Relative permeability$, \chi = Magnetic susceptibility Dimensionless	Non-ferromagnetic materials
Curie's Law	$$\chi = \frac{C}{T}$$	\chi = Susceptibility, C = Curie constant, T = Absolute temperature K	Weak fields, moderate temperatures
Curie Constant Expression	$$C = \frac{\mu_0 N \mu_m^2}{3 k_B}$$	N = Number density of dipoles, \m$u_m = Magnetic moment$, $k_B = Boltzmann constant$ K	Classical ideal paramagnet
Curie-Weiss Law	$$\chi = \frac{C}{T - \theta}$$	$\theta = Curie-Weiss temperature$ K	Temperatures near transition or interacting spins
Langevin Function	$$L(x) = \coth(x) - \frac{1}{x}$$	$x = \frac{\mu_m B}{k_B T}$ Dimensionless	Classical statistical mechanics derivation
Macroscopic Magnetic Induction	$$B = \mu_0 (H + M)$$	B = Magnetic induction, H = Magnetic field intensity, M = Magnetization Tesla (T)	General magnetic media
Effective Bohr Magneton Number	$$\mu_{eff} = g \sqrt{J(J+1)} \mu_B$$	$g = Lande g-factor$, J = Total angular momentum quantum number, \m$u_B = Bohr magneton$ Joule per Tesla (J/T) or Bohr magneton	Quantum theory of paramagnetism

Graphs and experiments

Susceptibility vs. Temperature (Curie's Law Graph)

Draw: A rectangular hyperbola plot of magnetic susceptibility (χ) on the y-axis versus absolute temperature (T) on the x-axis.

Interpret: Shows that susceptibility drops sharply as temperature increases, verifying inverse proportionality.

Exam use: Frequently asked in conceptual multiple-choice questions regarding the temperature dependence of paramagnetic materials.

Inverse Susceptibility vs. Temperature (Curie-Weiss Plot)

Draw: A straight line plot of 1/χ versus T intersecting the positive temperature axis at θ for ferromagnets turned paramagnets, or passing through the origin for pure paramagnets.

Interpret: The slope of the straight line equals 1/C, and the temperature intercept yields the Curie-Weiss temperature θ.

Exam use: Tested in PGT numericals to determine the Curie constant and Curie temperature from graphical data.

Quincke's Tube Method (Experimental Setup)

Draw: A U-tube containing a paramagnetic liquid placed between the pole pieces of an electromagnet such that one liquid meniscus is in a strong magnetic field region and the other is outside.

Interpret: The liquid level rises or falls due to magnetic force acting on the volume susceptibility of the fluid.

Exam use: Descriptive questions on experimental measurement of magnetic susceptibility of liquids.

Solved numericals

TGT core

A paramagnetic substance has a magnetic susceptibility of 3.0 × 10^-4 at a temperature of 300 K. Calculate its susceptibility at a temperature of 450 K.

Identify Curie's Law relation: χ₁ T₁ = χ₂ T₂.
Substitute the given values: (3.0 × 10^-4) × $300 = χ_{2} × 450$.
Solve for χ₂: χ₂ = (3.0 × 10^-4 × 300) / $450 = 9.0 × 10^{-2} / 450 = 2.0 × 10^{-4}$.

Answer: 2.0 × 10^-4

PGT extension

The magnetic susceptibility of a paramagnetic salt is 0.01 at 27°C. At what temperature will its susceptibility become 0.02 according to Curie's Law?

Convert initial temperature to Kelvin: T₁ = 27 + 273 = 300 K.
Apply Curie's Law: χ₁ T₁ = χ₂ T₂.
Substitute the given values: 0.01 × $300 = 0.02 × T_{2}$.
Calculate T₂: T₂ = (0.01 × 300) / 0.02 = 150 K.

Answer: 150 K (or -123°C)

PGT extension

A paramagnetic material follows the Curie-Weiss law with a Curie constant of 1.2 K and a Curie-Weiss temperature of 50 K. Calculate its magnetic susceptibility at 150 K.

Write down the Curie-Weiss law formula: χ = C / (T - θ).
Substitute C = 1.2 K, T = 150 K, and θ = 50 K.
Evaluate the denominator: T - θ = 150 - 50 = 100 K.
Calculate χ: χ = 1.2 / 100 = 0.012.

Answer: 0.012

Assumptions and limitations

Curie's law assumes independent magnetic dipoles without mutual interaction, which fails at very low temperatures or high field strengths where quantum saturation or cooperative magnetic ordering occurs.

Common misconceptions

Common error: Paramagnetic materials retain permanent magnetization after the external magnetic field is switched off.

Correct: Paramagnetic materials lose their magnetization completely when the external magnetic field is removed due to rapid thermal agitation.

Retaining permanent magnetization is a characteristic of ferromagnetism, not paramagnetism.

Common error: Magnetic susceptibility of a paramagnetic substance is negative.

Correct: Magnetic susceptibility of a paramagnetic substance is always small and positive.

Negative susceptibility is exclusively a characteristic of diamagnetic substances.

Quick revision

Paramagnetism is characterized by a small positive susceptibility (χ > 0) and inverse temperature dependence (χ ∝ 1/T).
Magnetic Susceptibility Definition: \chi = \frac{M}{H}
Relative Permeability: \m$u_r = 1 + \chi$

Must remember

Paramagnetic materials possess a net permanent magnetic dipole moment due to partially filled inner d or f electron shells.
Magnetic susceptibility $\chi$ of a paramagnetic substance is small and positive, typically ranging from $10^{-5}$ to $10^{-2}$.
Relative permeability $\mu_r$ is slightly greater than unity, given by $\mu_r = 1 + \chi$.
Curie's Law states that magnetic susceptibility is inversely proportional to absolute temperature: $\chi = C / T$.
The Curie-Weiss Law modifies Curie's law for interacting spins or lower temperatures: $\chi = C / (T - \theta)$.
Thermal agitation randomizes atomic dipoles in the absence of an external magnetic field, resulting in zero net magnetization.
Paramagnetic materials completely lose their magnetization when the external magnetic field is removed.
Negative magnetic susceptibility is a characteristic of diamagnetism, whereas paramagnetism is always positive.

Glossary

Magnetic Susceptibility ($\chi$)A dimensionless proportionality constant indicating the degree of magnetization of a material in response to an applied magnetic field.

Relative Permeability ($\mu_r$)The ratio of the permeability of a medium to the permeability of free space, given by $1 + \chi$.

Curie's Constant ($C$)A material-specific constant that appears in Curie's law of paramagnetism.

Curie-Weiss Temperature ($\theta$)A characteristic temperature representing the strength of molecular interactions in the Curie-Weiss law.

Bohr Magneton ($\mu_B$)A physical constant and natural unit for expressing the magnetic moment of an electron caused by its orbital or spin angular momentum.

Langevin Function ($L(x)$)A function arising in the classical statistical mechanical derivation of paramagnetism, defined as $\coth(x) - 1/x$.

Comparison of Magnetic Properties

Parameter	Paramagnetism	Diamagnetism
Susceptibility ($\chi$)	Small and positive ($10^{-5}$ to $10^{-2}$)	Small and negative ($-10^{-5}$ to $-10^{-9}$)
Relative Permeability ($\mu_r$)	Slightly greater than 1 ($> 1$)	Slightly less than 1 ($< 1$)
Effect of Temperature	Follows Curie's Law ($\chi \propto 1/T$)	Independent of temperature
Origin	Permanent atomic dipole moments	Induced dipole moments opposing field

Practice Quiz

Choose an option to reveal the explanation.

1. What is the typical range of magnetic susceptibility $\chi$ for a paramagnetic substance?

Explanation: Paramagnetic materials have a small and positive magnetic susceptibility, typically ranging between $10^{-5}$ and $10^{-2}$.

2. According to Curie's Law, how does the magnetic susceptibility $\chi$ of a paramagnetic substance vary with absolute temperature $T$?

Explanation: Curie's law states that $\chi = \frac{C}{T}$, meaning susceptibility is inversely proportional to absolute temperature.

3. What is the relation between relative permeability $\mu_r$ and magnetic susceptibility $\chi$?

Explanation: The relative permeability is defined by the relation $\mu_r = 1 + \chi$ for non-ferromagnetic materials.

4. Which of the following atomic configurations is primarily responsible for permanent atomic dipole moments in paramagnetic materials?

Explanation: Paramagnetic materials consist of atoms, ions, or molecules having net permanent magnetic dipole moments due to partially filled inner electron shells, such as d or f orbitals.

5. What happens to the magnetization of a paramagnetic material when the external magnetic field is switched off?

Explanation: Due to rapid thermal agitation, paramagnetic materials completely lose their macroscopic magnetization once the external magnetic field is removed.

6. How is the Curie-Weiss law expressed mathematically?

Explanation: The Curie-Weiss law is expressed as $\chi = \frac{C}{T - \theta}$, where $\theta$ is the Curie-Weiss temperature.

7. In the context of the macroscopic magnetic induction inside a paramagnetic material, what does the relation $B = \mu_0(H + M)$ signify?

Explanation: The magnetic induction $B$ inside a material accounts for both the applied magnetic field intensity $H$ and the material's induced magnetization $M$ via $B = \mu_0(H + M)$.

8. What is the Langevin function $L(x)$ used in the classical statistical mechanics derivation of paramagnetism?

Explanation: The Langevin function is defined as $L(x) = \coth(x) - \frac{1}{x}$, where $x = \frac{\mu_m B}{k_B T}$.

9. Why does Curie's law fail at temperatures very close to absolute zero?

Explanation: At temperatures close to absolute zero or in the presence of strong molecular interactions, Curie's law deviates, and the Curie-Weiss law or quantum treatments are required.

10. Which of the following statements about the magnetic susceptibility of paramagnets is correct?

Explanation: At moderate fields and temperatures, susceptibility is independent of the magnetic field intensity and inversely proportional to temperature.

11. What is the effective Bohr magneton number formula in the quantum theory of paramagnetism?

Explanation: In quantum theory, the effective magnetic moment is given by $\mu_{eff} = g \sqrt{J(J+1)} \mu_B$, where $g$ is the Lande g-factor and $J$ is the total angular momentum quantum number.

12. Which condition must be met for Curie's law to hold accurately?

Explanation: Curie's law holds well when thermal energy ($k_B T$) significantly exceeds the magnetic potential energy, corresponding to weak fields and moderate temperatures.

13. What is the unit of the Curie constant $C$ in the expression $\chi = \frac{C}{T}$?

Explanation: Since susceptibility $\chi$ is dimensionless and temperature $T$ is in Kelvin, the Curie constant $C$ has units of Kelvin (K).

14. Which physical parameter represents the number density of dipoles in the microscopic expression for the Curie constant?

Explanation: In the Curie constant expression $C = \frac{\mu_0 N \mu_m^2}{3 k_B}$, $N$ represents the number density of dipoles.

15. Can a paramagnetic substance be attracted into a region of a stronger magnetic field?

Explanation: Paramagnetic substances are weakly attracted towards regions of stronger magnetic field because their susceptibility is positive.

PYQ Patterns

These are pattern-based practice questions. Authentic year labels are only used when verified from a source.

1. The magnetic susceptibility of a paramagnetic material is found to be $0.005$ at $300\text{ K}$. What is its susceptibility at $600\text{ K}$ according to Curie's Law?

Explanation: According to Curie's law, $\chi \propto \frac{1}{T}$. Therefore, $\chi_2 = \chi_1 \times \frac{T_1}{T_2} = 0.005 \times \frac{300}{600} = 0.0025$.

2. If the relative permeability of a paramagnetic medium is $1.002$, what is its magnetic susceptibility?

Explanation: Using the relation $\mu_r = 1 + \chi$, we get $\chi = \mu_r - 1 = 1.002 - 1 = 0.002$.

3. In a Curie-Weiss plot of $\frac{1}{\chi}$ versus absolute temperature $T$ for a paramagnetic substance, what does the intercept on the temperature axis represent?

Explanation: From the Curie-Weiss law $\chi = \frac{C}{T - \theta}$, taking the reciprocal gives $\frac{1}{\chi} = \frac{T - \theta}{C}$, which intercepts the temperature axis at $T = \theta$.

4. Which of the following correctly describes the magnetic induction $B$ inside a paramagnetic rod placed in a magnetic field intensity $H$?

Explanation: Inside a paramagnetic material, the magnetization $M$ is positive and aligns with $H$, making $B = \mu_0(H + M)$ slightly larger than $\mu_0 H$.

5. What is the primary reason why paramagnetism is temperature-dependent whereas diamagnetism is largely temperature-independent?

Explanation: Paramagnetism depends on thermal competition with external torque aligning permanent dipoles, whereas diamagnetism arises from induced orbital current changes unaffected by thermal kinetic energy.

6. According to classical statistical mechanics, the Curie constant $C$ is proportional to which of the following quantities squared?

Explanation: The Curie constant expression $C = \frac{\mu_0 N \mu_m^2}{3 k_B}$ shows that $C$ is proportional to the square of the atomic magnetic moment $\mu_m$.

Timed Topic Test

Attempt these questions under a ten-minute limit using a separate timer.

1. If the absolute temperature of a paramagnetic substance is doubled while keeping the external magnetic field constant, its magnetization will approximately:

Explanation: For weak fields and moderate temperatures, magnetization $M \propto \chi \propto \frac{1}{T}$. Doubling $T$ halves the magnetization.

2. What is the value of magnetic susceptibility for a vacuum?

Explanation: Magnetic susceptibility $\chi$ is zero for a vacuum, as a vacuum has no matter to magnetize.

3. Which electronic shell type is most frequently associated with paramagnetic ions in transition metals and rare earths?

Explanation: Transition metals (d-shells) and rare-earth elements (f-shells) have partially filled inner shells containing unpaired electrons, giving rise to permanent net magnetic moments.

4. In the Curie-Weiss law, if a material has a positive Curie-Weiss temperature $\theta > 0$, what type of dominant cooperative interaction exists between atomic magnetic moments?

Explanation: A positive Curie-Weiss temperature ($\theta > 0$) typically indicates ferromagnetic interactions among the magnetic moments below the Curie point.

5. What is the dimension of magnetic susceptibility $\chi$?

Explanation: Magnetic susceptibility is defined as the ratio of magnetization $M$ to magnetic field intensity $H$, both having units of $\text{A}/\text{m}$, making $\chi$ dimensionless.

6. How does the magnetic induction $B$ inside a paramagnetic material compare to that in free space $B_0$ for the same field intensity $H$?

Explanation: Since $\mu_r > 1$ for paramagnets ($B = \mu_r B_0$), the magnetic induction inside the material is slightly larger than the vacuum magnetic field $B_0$.

7. Which of the following laws transitions into Curie's law when the Curie-Weiss temperature $\theta$ approaches zero?

Explanation: The Curie-Weiss law $\chi = \frac{C}{T - \theta}$ reduces directly to Curie's law $\chi = \frac{C}{T}$ when $\theta = 0$.

8. In quantum paramagnetism, what does the Lande g-factor ($g$) account for?

Explanation: The Lande g-factor relates the total angular momentum of an electron system to its magnetic dipole moment, accounting for combined spin and orbital contributions.

9. What is the physical significance of the parameter $x$ in the Langevin function $L(x) = \coth(x) - \frac{1}{x}$?

Explanation: The parameter $x = \frac{\mu_m B}{k_B T}$ represents the ratio of the magnetic potential energy of the dipole in the field to the thermal energy $k_B T$.

10. Which material property distinguishes paramagnetism from ferromagnetism?

Explanation: Paramagnetism exhibits linear response, small positive susceptibility, and no remanent magnetization or hysteresis, unlike ferromagnetism.

Paramagnetism

Scope and prerequisites

TGT core

PGT extension

Prerequisites

Origin of Paramagnetism

Magnetic Susceptibility and Permeability

Curie's Law of Paramagnetism

Curie-Weiss Law

Derivations

Classical Langevin Theory of Paramagnetism

Relation between Relative Permeability and Susceptibility

Formula sheet

Graphs and experiments

Susceptibility vs. Temperature (Curie's Law Graph)

Inverse Susceptibility vs. Temperature (Curie-Weiss Plot)

Quincke's Tube Method (Experimental Setup)

Solved numericals

A paramagnetic substance has a magnetic susceptibility of 3.0 × 10-4 at a temperature of 300 K. Calculate its susceptibility at a temperature of 450 K.

The magnetic susceptibility of a paramagnetic salt is 0.01 at 27°C. At what temperature will its susceptibility become 0.02 according to Curie's Law?

A paramagnetic material follows the Curie-Weiss law with a Curie constant of 1.2 K and a Curie-Weiss temperature of 50 K. Calculate its magnetic susceptibility at 150 K.

Assumptions and limitations

Common misconceptions

Quick revision

Must remember

Glossary

Comparison of Magnetic Properties

Practice Quiz

1. What is the typical range of magnetic susceptibility $\chi$ for a paramagnetic substance?

2. According to Curie's Law, how does the magnetic susceptibility $\chi$ of a paramagnetic substance vary with absolute temperature $T$?

3. What is the relation between relative permeability $\mu_r$ and magnetic susceptibility $\chi$?

4. Which of the following atomic configurations is primarily responsible for permanent atomic dipole moments in paramagnetic materials?

5. What happens to the magnetization of a paramagnetic material when the external magnetic field is switched off?

6. How is the Curie-Weiss law expressed mathematically?

7. In the context of the macroscopic magnetic induction inside a paramagnetic material, what does the relation $B = \mu_0(H + M)$ signify?

8. What is the Langevin function $L(x)$ used in the classical statistical mechanics derivation of paramagnetism?

9. Why does Curie's law fail at temperatures very close to absolute zero?

10. Which of the following statements about the magnetic susceptibility of paramagnets is correct?

11. What is the effective Bohr magneton number formula in the quantum theory of paramagnetism?

12. Which condition must be met for Curie's law to hold accurately?

13. What is the unit of the Curie constant $C$ in the expression $\chi = \frac{C}{T}$?

14. Which physical parameter represents the number density of dipoles in the microscopic expression for the Curie constant?

15. Can a paramagnetic substance be attracted into a region of a stronger magnetic field?

PYQ Patterns

1. The magnetic susceptibility of a paramagnetic material is found to be $0.005$ at $300\text{ K}$. What is its susceptibility at $600\text{ K}$ according to Curie's Law?

2. If the relative permeability of a paramagnetic medium is $1.002$, what is its magnetic susceptibility?

3. In a Curie-Weiss plot of $\frac{1}{\chi}$ versus absolute temperature $T$ for a paramagnetic substance, what does the intercept on the temperature axis represent?

4. Which of the following correctly describes the magnetic induction $B$ inside a paramagnetic rod placed in a magnetic field intensity $H$?

5. What is the primary reason why paramagnetism is temperature-dependent whereas diamagnetism is largely temperature-independent?

6. According to classical statistical mechanics, the Curie constant $C$ is proportional to which of the following quantities squared?

Timed Topic Test

1. If the absolute temperature of a paramagnetic substance is doubled while keeping the external magnetic field constant, its magnetization will approximately:

2. What is the value of magnetic susceptibility for a vacuum?

3. Which electronic shell type is most frequently associated with paramagnetic ions in transition metals and rare earths?

4. In the Curie-Weiss law, if a material has a positive Curie-Weiss temperature $\theta > 0$, what type of dominant cooperative interaction exists between atomic magnetic moments?

5. What is the dimension of magnetic susceptibility $\chi$?

6. How does the magnetic induction $B$ inside a paramagnetic material compare to that in free space $B_0$ for the same field intensity $H$?

7. Which of the following laws transitions into Curie's law when the Curie-Weiss temperature $\theta$ approaches zero?

8. In quantum paramagnetism, what does the Lande g-factor ($g$) account for?

9. What is the physical significance of the parameter $x$ in the Langevin function $L(x) = \coth(x) - \frac{1}{x}$?

10. Which material property distinguishes paramagnetism from ferromagnetism?

A paramagnetic substance has a magnetic susceptibility of 3.0 × 10^-4 at a temperature of 300 K. Calculate its susceptibility at a temperature of 450 K.