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Electricity and Magnetism
Ferromagnetism
Ferromagnetism is a phenomenon where certain materials, such as iron, cobalt, and nickel, exhibit strong permanent magnetization below a specific characteristic temperature known as the Curie temperature. This behavior arises from the strong quantum mechanical exchange coupling between adjacent atomic magnetic dipole moments, aligning them parallel to each other within microscopic regions called domains.
UP TGT ScienceUP PGT PhysicsStudy Notes + Practice
Scope and prerequisites
TGT core
Qualitative understanding of ferromagnetic substances, domain theory, Curie temperature, hysteresis loop, and basic retentivity and coercivity concepts.
PGT extension
Quantitative treatment of Weiss molecular field theory, Curie-Weiss law for susceptibility, magnetic domain wall energy, and energy losses in hysteresis cycles.
Prerequisites
Atomic magnetic dipole moments and Bohr magneton
Magnetic intensity (H), magnetic induction (B), and magnetization (M)
Paramagnetism and diamagnetism fundamentals
Origin of Ferromagnetism and Exchange Interaction
Ferromagnetism originates from the quantum mechanical exchange interaction, electrostatic in origin, which couples adjacent electron spins.
The exchange energy favors parallel alignment of spins when the exchange integral is positive, overcoming thermal agitation at low temperatures.
It leads to spontaneous magnetization within small regions termed magnetic domains even in the absence of an external magnetic field.
Domain Theory of Ferromagnetism
A macroscopic ferromagnetic sample is divided into numerous microscopic regions called domains, each uniformly magnetized to saturation.
In an unmagnetized state, the net magnetic moment of the bulk material is zero due to the random orientations of the individual domain magnetic moments.
Application of an external magnetic field causes domain growth through boundary displacement and subsequent domain rotation toward the field direction.
Curie-Weiss Law and Temperature Dependence
Above a critical temperature called the Curie temperature (TC), a ferromagnetic material transitions into a paramagnetic state.
The magnetic susceptibility (χ) above TC follows the Curie-Weiss law given by $\chi = C / (T - T_{C})$, where C is the Curie constant.
Below TC, the thermal energy is insufficient to overcome the internal exchange field, preserving spontaneous alignment.
Hysteresis and Magnetic Properties
Hysteresis is the lag of magnetic induction (B) behind the magnetizing field (H) during a complete cycle of magnetization and demagnetization.
Retentivity (or remanence) is the residual magnetic induction retained by the material when the external field H is reduced to zero.
Coercivity is the magnitude of the reverse magnetizing field required to reduce the residual magnetic induction to zero.
Derivations
Weiss Molecular Field Theory and Curie-Weiss Law
Conditions: Applied to ferromagnetic materials in the paramagnetic region (T > T_C)
Assume an internal molecular field Bm proportional to the magnetization M, given by Bm = λM, where λ is the Weiss molecular field constant.
The total effective magnetic field acting on the atomic dipoles is Beff = B + Bm = B + λM, where B is the applied external field.
Using the standard paramagnetic susceptibility formula based on Langevin's theory, $M = (C' / T) B_{eff}$, substitute Beff to get $M = (C' / T)(B + \lambdaM)$.
Express B / μ0 as H and use $M = \chiH$. Rearrange the equation to solve for magnetic susceptibility χ.
Result: $\chi = \frac{C}{T - T_C} \text{}$ where } $T_C = \lambda C' \mu_0$
Energy Dissipation per Unit Volume per Hysteresis Cycle
Conditions: Quasi-static cyclic magnetization of a ferromagnetic core
The work done per unit volume in changing the magnetization by dM in an external field H is dW = H dB.
Since $B = \mu_0 (H + M)$, the infinitesimal work is d$W = \mu_0 H dH + \mu_0 H dM$.
Integrating over a complete closed hysteresis cycle, the term involving H dH vanishes because the initial and final states are identical.
The net work done, which appears entirely as thermal energy dissipated per unit volume per cycle, equals the area enclosed by the B-H (or M-H) hysteresis loop.
Result: W = \oint H \, d$B = \text{Area of the B-H loop}$
Formula sheet
Formula / law
Equation
Variables and units
Conditions
Curie-Weiss Law
$$\chi = \frac{C}{T - T_C}$$
$\chi = Magnetic susceptibility$, C = Curie constant, T = Absolute temperature, TC = Curie temperature Dimensionless (χ), K (T)
Valid for T > TC
Magnetic Induction in Ferromagnet
$$B = \mu_0 (H + M)$$
B = Magnetic induction, μ0 = Permeability of free space, H = Magnetic intensity, M = Magnetization Tesla (T), A/m (H, M)
General macroscopic relation
Relative Permeability
$$\mu_r = 1 + \chi$$
μr = Relative magnetic permeability, $\chi = Magnetic susceptibility$ Dimensionless
Linear isotropic media approximation
Energy Loss per Cycle
$$E = \oint H \, dB$$
E = Energy dissipated per unit volume per cycle, H = Magnetic field intensity, B = Magnetic induction Joule per cubic meter (J/m3)
Closed thermodynamic cycle
Bohr Magneton
$$\mu_B = \frac{e \hbar}{2 m_e}$$
e = Elementary charge, $\hbar = Reduced Planck constant$, $m_e = Mass of electron$ A·m2 or J/T
Fundamental atomic magnetic moment unit
Steinmetz Hysteresis Loss Formula
$$W_h = η B_{max}^{1.6}$$
Wh = Hysteresis energy loss per cycle, $\eta = Steinmetz coefficient$, Bmax = Maximum magnetic induction J/m3/cycle
Empirical relation for electrical steels
Magnetization Vector Relation
$$M = \frac{\sum \vec{m}}{V}$$
∑m = Vector sum of magnetic dipole moments, V = Volume A/m
Macro-scale uniform averaging
Weiss Molecular Field
$$B_m = \lambda M$$
Bm = Molecular field, $\lambda = Weiss field parameter$, M = Magnetization Tesla (T)
Mean field approximation
Graphs and experiments
B-H Hysteresis Loop
Draw: A closed loop plotting Magnetic Induction (B) on the y-axis against Magnetic Intensity (H) on the x-axis, showing saturation, retentivity (Br), and coercivity (Hc).
Interpret: The non-linear response and loop area quantify the energy dissipated per cycle. Retentivity indicates memory retention; coercivity indicates resistance to demagnetization.
Exam use: Direct identification questions on soft and hard magnetic materials based on loop geometry (narrow vs. wide).
Susceptibility vs. Temperature (Curie-Weiss Law)
Draw: A plot of inverse magnetic susceptibility (1/χ) versus temperature (T), showing a linear increase above TC that extrapolates to intercept the temperature axis at $T = T_{C}$.
Interpret: Demonstrates the phase transition from ferromagnetism to paramagnetism at the Curie point.
Exam use: Numerical and conceptual problems on Curie constant calculation and temperature scaling.
Rowland Ring Experiment
Draw: A toroidal ferromagnetic ring wound uniformly with a primary magnetizing coil and a secondary search coil connected to a ballistic galvanometer.
Interpret: Allows experimental determination of the B-H curve and hysteresis loops for a given ferromagnetic specimen.
Exam use: Conceptual questions on experimental setup parameters and flux linkage measurement.
Solved numericals
TGT core
A ferromagnetic material has a relative permeability of 1000. Calculate its magnetic susceptibility.
Recall the relation between relative permeability (μr) and magnetic susceptibility (χ): μr = 1 + χ.
Rearrange to solve for susceptibility: $\chi = \mu_{r} - 1$.
Substitute the given value: $\chi = 1000 - 1 = 999$.
Answer: 999 (Dimensionless)
PGT extension
The magnetic susceptibility of a ferromagnetic material is 4.0 × 10-3 at a temperature of 400 K. If its Curie temperature is 300 K, find the susceptibility at 500 K.
According to the Curie-Weiss law, $\chi = C / (T - T_{C})$, which implies χ(T - TC) = C (a constant).
Set up the ratio for two temperatures T1 = 400 K and T2 = 500 K with TC = 300 K: χ1(T1 - TC) = χ2(T2 - TC).
Calculate the energy dissipated per unit volume per cycle for a hysteresis loop whose area is equivalent to 500 J/m3.
Recall that the energy loss per unit volume per cycle is numerically equal to the area of the B-H (or μ0H-M) hysteresis loop.
Given the loop area directly represents the work done per unit volume: $W = \oint H dB = 500 J/m^{3}$.
Answer: 500 J/m3
Assumptions and limitations
Weiss molecular field theory assumes a uniform effective internal field acting on every atomic dipole, which is an oversimplification of quantum mechanical exchange interactions.
Common misconceptions
Common error: Ferromagnetic materials retain permanent magnetism indefinitely without any external influence.
Correct: Permanent magnetism depends on high coercivity and retentivity; mechanical shock, heating above TC, or stray reverse fields can demagnetize them.
Thermal agitation and stray magnetic fields can disrupt domain alignment over time if coercivity is low.
Common error: Magnetic susceptibility of a ferromagnet is constant like that of a paramagnet.
Correct: Susceptibility in ferromagnets is highly non-linear, field-dependent, and varies strongly with temperature according to the Curie-Weiss law above TC.
Ferromagnetism involves cooperative domain phenomena rather than independent atomic dipole responses.
Quick revision
Ferromagnetism arises from quantum exchange coupling aligning atomic dipoles parallel within domains.
Above the Curie temperature (TC), a ferromagnet becomes paramagnetic obeying $\chi = C / (T - T_{C})$.
Area of the B-H hysteresis loop represents thermal energy loss per unit volume per cycle.
Soft magnetic materials have narrow loops (low coercivity, high permeability), while hard magnetic materials have wide loops (high coercivity, high retentivity).
Must remember
Ferromagnetism originates from the quantum mechanical exchange interaction that couples adjacent electron spins.
Above the Curie temperature ($T_C$), a ferromagnetic substance transforms into a paramagnetic substance obeying the Curie-Weiss law $\chi = \frac{C}{T - T_C}$.
Magnetic hysteresis represents the lag of magnetic induction ($B$) behind the magnetizing field ($H$) during a full cycle.
Retentivity is the residual magnetism when $H = 0$, while coercivity is the reverse field required to reduce $B$ to zero.
The Weiss molecular field approximation expresses the internal effective field as $B_m = \lambda M$.
Glossary
Exchange InteractionA quantum mechanical electrostatic interaction that aligns adjacent electron spins parallel or antiparallel.
Curie Temperature ($T_C$)The critical temperature above which a ferromagnetic material loses spontaneous magnetization and becomes paramagnetic.
RetentivityThe ability of a material to retain residual magnetization when the external magnetizing field is removed.
CoercivityThe magnitude of the reverse magnetic field needed to bring residual magnetic induction down to zero.
Weiss Molecular FieldAn effective internal magnetic field assumed by Weiss to align atomic dipoles in ferromagnets, proportional to magnetization.
Comparison of Magnetic States
Property
Ferromagnetism
Paramagnetism
Susceptibility ($\chi$)
Large, positive, and field-dependent
Small, positive, and temperature-dependent
Temperature Dependence
Follows Curie-Weiss law ($\chi = \frac{C}{T - T_C}$) above $T_C$
Follows Curie law ($\chi = \frac{C}{T}$)
Spontaneous Magnetization
Present below $T_C$ due to domains
Absent
Practice Quiz
Choose an option to reveal the explanation.
1. What is the primary physical origin of ferromagnetism in solid materials?
Explanation: Ferromagnetism fundamentally originates from the quantum mechanical exchange interaction, which is electrostatic in origin and favors parallel spin alignments.
2. How does the magnetic susceptibility ($\chi$) of a ferromagnetic material vary with temperature above its Curie temperature ($T_C$)?
Explanation: Above $T_C$, ferromagnetic materials become paramagnetic and follow the Curie-Weiss law given by $\chi = \frac{C}{T - T_C}$.
3. In an unmagnetized ferromagnetic sample, what is the net magnetic moment of the bulk material?
Explanation: In an unmagnetized state, the individual domains are randomly oriented, resulting in a net macroscopic magnetic moment of zero.
4. What physical parameter does retentivity represent in a hysteresis loop ($B-H$ curve)?
Explanation: Retentivity (or remanence) is the residual magnetic induction retained by the material when the external magnetizing field $H$ is removed.
5. Which of the following defines coercivity?
Explanation: Coercivity is specifically the magnitude of the reverse magnetizing field needed to eliminate the residual magnetic induction.
6. What does the area enclosed by a magnetic hysteresis loop represent?
Explanation: The area of the $B-H$ hysteresis loop represents the thermal energy dissipated per unit volume per complete cycle of magnetization and demagnetization ($E = \oint H \, dB$).
7. According to the Weiss molecular field theory, what is the relationship between the molecular field $B_m$ and magnetization $M$?
Explanation: The Weiss molecular field is proportional to the magnetization, expressed as $B_m = \lambda M$, where $\lambda$ is the Weiss field parameter.
8. What happens to domains under the application of an external magnetic field?
Explanation: An external field causes favorable domains to grow via domain wall boundary displacement followed by rotation of magnetic moments toward the field direction.
9. Which unit is appropriate for magnetic induction $B$?
Explanation: Magnetic induction $B$ is measured in Teslas (T), whereas magnetic intensity $H$ and magnetization $M$ are measured in A/m.
10. What is the relation between relative permeability $\mu_r$ and magnetic susceptibility $\chi$ in linear isotropic media?
Explanation: The relative permeability is related to susceptibility by the standard constitutive relation $\mu_r = 1 + \chi$.
11. What is the physical significance of the Bohr magneton ($\mu_B$)?
Explanation: The Bohr magneton $\mu_B = \frac{e \hbar}{2 m_e}$ serves as the natural atomic unit for expressing magnetic dipole moments of electrons.
12. Steinmetz's empirical formula for hysteresis energy loss per cycle is given by $W_h = \eta B_{\max}^{1.6}$. What does $\eta$ represent?
Explanation: In Steinmetz formula, $\eta$ is the Steinmetz coefficient which characterizes the specific magnetic material's hysteresis loss intensity.
13. Why do ferromagnetic materials lose their permanent magnetism when heated above the Curie temperature?
Explanation: Thermal agitation at temperatures above $T_C$ disrupts the cooperative parallel spin alignment enforced by the exchange interaction, randomizing the atomic moments.
14. Which macroscopic relation correctly connects magnetic induction $B$, magnetic intensity $H$, and magnetization $M$?
Explanation: The general macroscopic relation in magnetic media is $B = \mu_0 (H + M)$.
15. Which of the following statements correctly contrasts ferromagnetism with paramagnetism regarding susceptibility?
Explanation: Ferromagnetic susceptibility is highly non-linear and depends strongly on the magnetic field and domain configurations, unlike simple paramagnetism.
PYQ Patterns
These are pattern-based practice questions. Authentic year labels are only used when verified from a source.
1. If the Curie temperature of iron is about $1043\text{ K}$, what is its magnetic behavior at $1200\text{ K}$?
Explanation: When temperature exceeds the Curie temperature ($T > T_C$), a ferromagnetic material undergoes a phase transition into a paramagnetic state.
2. The magnetic susceptibility of a ferromagnetic material tested at a temperature $T$ well above $T_C$ is proportional to:
Explanation: According to the Curie-Weiss law, $\chi = \frac{C}{T - T_C}$, making susceptibility proportional to $\frac{1}{T - T_C}$.
3. In a hysteresis loop, the value of magnetic field $H$ required to reduce the residual induction to zero is called:
Explanation: Coercivity is defined as the reverse magnetic field intensity needed to reduce the remaining magnetic induction to zero.
4. Which of the following equations correctly expresses the Bohr magneton $\mu_B$?
Explanation: The fundamental atomic magnetic moment unit, the Bohr magneton, is given by $\mu_B = \frac{e \hbar}{2 m_e}$ where $e$ is elementary charge, $\hbar$ is reduced Planck constant, and $m_e$ is electron mass.
5. What physical entity is responsible for coupling adjacent electron spins to favor parallel alignment in ferromagnets?
Explanation: The exchange interaction, which is quantum mechanical and electrostatic in origin, couples adjacent spins to favor parallel alignment when the exchange integral is positive.
6. Soft iron is preferred for making electromagnets because it has:
Explanation: Soft iron exhibits high magnetic permeability, high retentivity, but very low coercivity and narrow hysteresis loop, allowing easy magnetization and demagnetization with minimal energy loss.
Timed Topic Test
Attempt these questions under a ten-minute limit using a separate timer.
1. What is the primary condition for the exchange integral to favor parallel alignment of spins in ferromagnetism?
Explanation: A positive exchange integral results in a lower energy state when adjacent electron spins are aligned parallel to each other.
2. If a ferromagnetic sample has a Curie temperature of $770^{\circ}\text{C}$, what is this temperature in Kelvin?
3. Which of the following represents the correct unit of the Weiss field parameter $\lambda$ in $B_m = \lambda M$?
Explanation: Since $B_m$ is in Tesla and $M$ is in A/m, $\lambda$ must have units of $\text{T}\cdot\text{m}/\text{A}$ (or dimensionless depending on definition conventions, but dimensionally equivalent to $\mu_0$). Specifically, $\lambda$ is dimensionless in certain rationalized CGS forms or has units of $\mu_0$ equivalent.
4. During the magnetization of a ferromagnetic material, the domain walls shift. This boundary displacement is primarily driven by:
Explanation: Domain wall displacement occurs because domains whose magnetic moments are oriented closer to the external field direction grow at the expense of unfavorable domains, minimizing overall magnetic potential energy.
5. What does the Steinmetz hysteresis loss coefficient measure across different materials?
Explanation: Steinmetz coefficient $\eta$ scales the hysteresis energy loss per cycle ($W_h = \eta B_{\max}^{1.6}$) for specific magnetic steels.
6. What is the behavior of magnetic domains in a saturated ferromagnetic sample?
Explanation: At magnetic saturation, all microscopic domains are fully aligned or rotated in the direction of the applied magnetic field.
7. If relative permeability $\mu_r$ of a substance is slightly greater than 1 ($1.0000$ to few decimals), can it be classified as ferromagnetic?
Explanation: Ferromagnetic materials exhibit very large relative permeabilities ($\mu_r \gg 1$), whereas small values slightly greater than 1 characterize paramagnetism.
8. Which of the following correctly describes the susceptibility value range for a ferromagnetic material below its Curie temperature?
Explanation: Ferromagnetic susceptibility is positive and extremely large, often ranging from hundreds to tens of thousands depending on the magnetic field.
9. What limits the continuous increase of magnetization $M$ in a ferromagnet when a high external field $H$ is applied?
Explanation: Magnetization reaches a ceiling (saturation magnetization $M_s$) when all available magnetic dipoles within the material are completely aligned with the external field.
10. How does mechanical shock or severe vibration affect a permanent magnet made of ferromagnetic material?
Explanation: Mechanical shocks supply localized energy that can overcome coercivity barriers, disrupting domain alignment and reducing net residual magnetization.