A steel object can retain magnetization after the applied field is removed, while another specimen responds much more reversibly. The difference lives in how microscopic moments cooperate and rearrange. We will connect that small-scale picture to fields and curves you can measure.
We start from a current-loop moment, keep B, H and M distinct in SI, and use a constructed hysteresis cycle to connect intercepts and enclosed area to physical measurements.
What you will learn to do
- Distinguish a magnetic moment from magnetization and flux density.
- Predict dipole torque and distinguish uniform-field rotation from translation.
- Use linear susceptibility only within its stated regime.
- Explain why local magnetic order need not give a large bulk moment.
- Solve a simple demagnetizing-field model self-consistently.
- Read B–H intercepts and calculate cyclic loss with correct units.
Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Magnetic moments, materials and hysteresis Learning goals: - Distinguish a magnetic moment from magnetization and flux density. - Predict dipole torque and distinguish uniform-field rotation from translation. - Use linear susceptibility only within its stated regime. - Explain why local magnetic order need not give a large bulk moment. - Solve a simple demagnetizing-field model self-consistently. - Read B–H intercepts and calculate cyclic loss with correct units. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
BEFORE YOU READ
Bring back one idea
Try a sentence or a sketch before opening the comparison. If this is new, use the explanation to get started.
Can a collection of nonzero magnetic moments have zero total moment?
Compare your reasoning
Yes. Their vector contributions can cancel. Near-zero bulk magnetization does not by itself show that every microscopic magnetic contribution has vanished.
Revisit the explanation →01 / Magnetic moments, materials and hysteresis
Begin with magnetic moments, then average over matter
A current loop produces a magnetic field and experiences a torque in another field. Its magnetic moment packages the loop’s current, area and orientation into a vector. Atomic-scale orbital motion and intrinsic spin also contribute magnetic moments, although electron spin is not a literal tiny charged ball rotating about its axis.
A piece of matter contains many such contributions. Their vector sum may be small even when the individual moments are not. Thermal disorder, induced changes of electronic motion, interactions between moments and the way domains arrange themselves all affect the net response.
Magnetization M is magnetic moment per unit volume after averaging over a region large compared with atomic spacing. Do not confuse M with a single moment or with the magnetic flux density B. Keeping their units distinct prevents a large class of formula errors.
For identical coplanar loop turns, μ has units A·m² and points along the right-hand-rule normal. M has units A/m. The symbol μ for a moment here is distinct from a material permeability.
WORKED EXAMPLE
Convert a winding into a moment before calculating a torque
A small 500-turn coil carries 0.20 A and each turn encloses 4.0 cm². Find its magnetic moment magnitude.
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Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Magnetic moments, materials and hysteresis
Focus: Begin with magnetic moments, then average over matter
Learning goals:
- Distinguish a magnetic moment from magnetization and flux density.
- Predict dipole torque and distinguish uniform-field rotation from translation.
- Use linear susceptibility only within its stated regime.
- Explain why local magnetic order need not give a large bulk moment.
- Solve a simple demagnetizing-field model self-consistently.
- Read B–H intercepts and calculate cyclic loss with correct units.
Equations in this section (LaTeX):
\boldsymbol\mu=NI\boldsymbol A,\qquad \boldsymbol M=\frac{\text{net magnetic moment}}{\text{volume}}
Meaning: For identical coplanar loop turns, μ has units A·m² and points along the right-hand-rule normal. M has units A/m. The symbol μ for a moment here is distinct from a material permeability.
Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.
Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.02 / Magnetic moments, materials and hysteresis
A uniform field can turn a dipole without translating it
For a small fixed magnetic moment in a uniform externally imposed field, opposite forces can form a couple. Their resultant force is zero while their torque is not. The dipole rotates toward lower interaction energy.
The fixed-moment interaction energy is −μ·B. It is least when the moment is parallel to the field and greatest when antiparallel. Those orientations both have zero torque, so “zero torque” alone cannot distinguish stable from unstable equilibrium.
A translational force requires a spatially varying field for this ideal point-dipole model. Material attraction is therefore not explained by torque alone. Later, when a material’s moment is itself induced by the field, the reversible energy includes the work of building that moment; do not automatically reuse the fixed-moment expression unchanged.
The moment is assumed fixed in magnitude and small enough that the external field is effectively uniform over it. θ is the angle between moment and field, not the angle of the coil’s plane.
WORKED EXAMPLE
Distinguish torque, force and the energy barrier
The 0.040 A·m² moment lies in a uniform 0.15 T field at 30° to the field. Find torque magnitude and the energy increase in rotating it quasistatically from parallel to antiparallel.
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Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Magnetic moments, materials and hysteresis Focus: A uniform field can turn a dipole without translating it Learning goals: - Distinguish a magnetic moment from magnetization and flux density. - Predict dipole torque and distinguish uniform-field rotation from translation. - Use linear susceptibility only within its stated regime. - Explain why local magnetic order need not give a large bulk moment. - Solve a simple demagnetizing-field model self-consistently. - Read B–H intercepts and calculate cyclic loss with correct units. Equations in this section (LaTeX): \boldsymbol\tau=\boldsymbol\mu\times\boldsymbol B,\qquad U=-\mu B\cos\theta Meaning: The moment is assumed fixed in magnitude and small enough that the external field is effectively uniform over it. θ is the angle between moment and field, not the angle of the coil’s plane. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
03 / Magnetic moments, materials and hysteresis
Separate B, H and M in SI units
The macroscopic flux density B includes the response of the material as well as the contribution associated with externally supplied currents. In SI, H is an auxiliary magnetic field strength that allows free-current and material-response bookkeeping to be separated. The identity B = μ₀(H + M) relates the three averaged fields.
A linear, isotropic material can be described by M = χH over a stated range. Then B = μ₀(1 + χ)H = μH. This is a constitutive model, not a universal relation with one constant χ for every magnetic material, field and temperature.
H and M are measured in A/m, B in tesla, χ and relative permeability μr in dimensionless units. In other unit systems the same letters can occur with different factors. Our equations use SI throughout. Never combine a Gaussian-unit susceptibility formula with an SI permeability equation just because the symbols look familiar.
The first relation defines the macroscopic SI separation. The last two require a linear isotropic response using the internal H; they need not describe a ferromagnet over a complete cycle.
WORKED EXAMPLE
Find the field inside an ideal long magnetized region
An ideal long solenoid has 1200 turns per metre and carries 0.50 A. A homogeneous linear core has χ = 0.20; neglect end and demagnetizing effects. Find H, M and B.
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Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Magnetic moments, materials and hysteresis Focus: Separate B, H and M in SI units Learning goals: - Distinguish a magnetic moment from magnetization and flux density. - Predict dipole torque and distinguish uniform-field rotation from translation. - Use linear susceptibility only within its stated regime. - Explain why local magnetic order need not give a large bulk moment. - Solve a simple demagnetizing-field model self-consistently. - Read B–H intercepts and calculate cyclic loss with correct units. Equations in this section (LaTeX): \boldsymbol B=\mu_0(\boldsymbol H+\boldsymbol M),\quad \boldsymbol M=\chi\boldsymbol H,\quad \mu_r=1+\chi Meaning: The first relation defines the macroscopic SI separation. The last two require a linear isotropic response using the internal H; they need not describe a ferromagnet over a complete cycle. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
04 / Magnetic moments, materials and hysteresis
The sign of susceptibility is only a starting classification
In a diamagnetic response, the induced magnetization opposes the internal field, giving negative susceptibility in a simple linear description. In a paramagnetic response, net alignment produces a positive susceptibility. Small susceptibility does not mean that microscopic magnetic moments must be absent.
For a dilute set of approximately noninteracting permanent moments in a weak field, thermal disorder competes with alignment and yields the Curie form χ = C/T. Temperature is absolute and the weak-field condition matters. Not all positive susceptibility follows this simple law; electron conduction contributions can behave differently.
Ferromagnetic ordering involves cooperative interactions. Below a Curie temperature, regions can possess spontaneous magnetization without an applied field. The response is nonlinear and depends on magnetic history, so one constant positive χ and a universal C/T law are insufficient. A bulk specimen can still have near-zero net moment because different domains point in different directions.
| Response | Simple sign/behaviour | What must still be checked |
|---|---|---|
| Diamagnetic | Negative weak-field χ in a simple linear model | Geometry, field range and additional magnetic contributions |
| Paramagnetic | Positive response to the field | Curie’s C/T law needs appropriate moment and temperature assumptions |
| Ferromagnetic | Cooperative order, domains, saturation and history | A constant χ does not describe the whole magnetization cycle |
WORKED EXAMPLE
Apply a temperature law only after naming its regime
A dilute weak-field paramagnet follows χ = C/T over the range considered. Its susceptibility is 2.4 × 10⁻³ at 300 K. What does this model predict at 400 K?
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STOP & THINK
A ferromagnetic specimen has nearly zero net external moment. Which explanation can be consistent with local magnetic order?
- No microscopic moments can exist
- Different magnetized domains can largely cancel in the bulk sum
- Every atom must be above the Curie temperature
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Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Magnetic moments, materials and hysteresis Focus: The sign of susceptibility is only a starting classification Learning goals: - Distinguish a magnetic moment from magnetization and flux density. - Predict dipole torque and distinguish uniform-field rotation from translation. - Use linear susceptibility only within its stated regime. - Explain why local magnetic order need not give a large bulk moment. - Solve a simple demagnetizing-field model self-consistently. - Read B–H intercepts and calculate cyclic loss with correct units. Equations in this section (LaTeX): Build the physical reasoning before choosing an equation. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
05 / Magnetic moments, materials and hysteresis
The internal field depends on the shape of the specimen
An applied field is the field that would be produced by external sources without the specimen’s magnetization contribution. The internal H inside a finite magnetized body can differ because the body’s own magnetization produces a demagnetizing field.
For a uniformly magnetized ellipsoid along one principal axis, a scalar demagnetizing factor D gives Hint = Happ − DM. This simple form is geometry-specific; the corresponding field in an arbitrary shape can be nonuniform. For a sphere D = 1/3 in this SI convention.
Combining this geometry with M = χHint produces a self-consistency equation. It explains why inserting the applied H directly into the constitutive relation can overestimate M. It also separates a material parameter χ from the response of a particular shaped sample.
This expression assumes a uniform linear isotropic specimen in the ellipsoidal geometry. It is not a saturation or hysteresis model.
WORKED EXAMPLE
Solve for the field that the material actually experiences
A hypothetical linear sphere has χ = 3.0 and is placed in an applied field strength of 100 A/m. With D = 1/3, determine its internal H and M.
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Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Magnetic moments, materials and hysteresis
Focus: The internal field depends on the shape of the specimen
Learning goals:
- Distinguish a magnetic moment from magnetization and flux density.
- Predict dipole torque and distinguish uniform-field rotation from translation.
- Use linear susceptibility only within its stated regime.
- Explain why local magnetic order need not give a large bulk moment.
- Solve a simple demagnetizing-field model self-consistently.
- Read B–H intercepts and calculate cyclic loss with correct units.
Equations in this section (LaTeX):
H_{\mathrm{int}}=H_{\mathrm{app}}-DM,\qquad M=\frac{\chi}{1+D\chi}H_{\mathrm{app}}
Meaning: This expression assumes a uniform linear isotropic specimen in the ellipsoidal geometry. It is not a saturation or hysteresis model.
Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.
Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.06 / Magnetic moments, materials and hysteresis
A hysteresis loop records a material’s path, not one function B(H)
As a ferromagnetic core is cycled, domain walls move and moments rotate. Some changes are reversible and some require overcoming barriers. The field on the way back can therefore produce a different B from the same H on the outward sweep. A single straight line B = μH cannot represent that history.
Remanence is a retained value at zero H after a specified previous magnetization path. On a B–H loop, the intercept where B becomes zero defines a B-coercive field. A coercive field defined by M = 0 is a different convention and need not have exactly the same value. Always read which vertical quantity is plotted.
The constructed loop below is a teaching dataset, not a measured material curve. Each branch is parameterized by increasing H for readability, but its label states the actual sweep direction. Start at the negative endpoint, follow the increasing-H branch to the positive endpoint, then return on the decreasing-H branch.
Read the history at the same internal H field
- Increasing H
- -0.6
- Decreasing H
- -0.6
WORKED EXAMPLE
Read remanence and the B-coercive field
Use the constructed cycle. At H = 0, the increasing and decreasing branches have B = −0.20 T and +0.20 T. The zero-B crossings occur at +100 and −100 A/m. What remanence and coercivity magnitudes does this report?
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Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Magnetic moments, materials and hysteresis Focus: A hysteresis loop records a material’s path, not one function B(H) Learning goals: - Distinguish a magnetic moment from magnetization and flux density. - Predict dipole torque and distinguish uniform-field rotation from translation. - Use linear susceptibility only within its stated regime. - Explain why local magnetic order need not give a large bulk moment. - Solve a simple demagnetizing-field model self-consistently. - Read B–H intercepts and calculate cyclic loss with correct units. Equations in this section (LaTeX): Build the physical reasoning before choosing an equation. Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding. Tutoring approach: Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.
07 / Magnetic moments, materials and hysteresis
The area of the loop is an energy density per cycle
For a quasistatic magnetic cycle in a uniform sample, the field work per unit volume can be written as the loop integral of H dB. Its positive dissipated magnitude equals the enclosed area of the B–H loop, with units (A/m)·T = J/m³.
If a closed cycle is sampled as two B values at the same H coordinates, integrate their difference over H. For straight interpolation between points, the trapezoidal sum gives the exact area of that constructed polygon. For a measured curve, it is an approximation whose quality depends on sample spacing.
Multiply energy density by core volume to obtain energy per cycle and then by cycles per second to obtain power. This is the hysteretic contribution represented by the specified loop; additional dynamic or eddy-current losses require their own information. A narrow loop and an electrically resistive or laminated core address different sources of loss.
Use H in A/m, B in tesla and volume in m³. The sign of an oriented integral depends on traversal; dissipated energy is reported as a positive magnitude.
WORKED EXAMPLE
Turn the constructed loop into a core-loss estimate
The constructed loop has a trapezoidal area of 202 J/m³. If a uniform core of volume 20 cm³ follows this same quasistatic loop 50 times per second, what hysteresis power follows?
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Study with a tutor prompt
Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.
Topic: Magnetic moments, materials and hysteresis
Focus: The area of the loop is an energy density per cycle
Learning goals:
- Distinguish a magnetic moment from magnetization and flux density.
- Predict dipole torque and distinguish uniform-field rotation from translation.
- Use linear susceptibility only within its stated regime.
- Explain why local magnetic order need not give a large bulk moment.
- Solve a simple demagnetizing-field model self-consistently.
- Read B–H intercepts and calculate cyclic loss with correct units.
Equations in this section (LaTeX):
w_{\mathrm{cycle}}=\left|\oint H\,dB\right|=\text{loop area},\qquad P_{\mathrm{hyst}}=fVw_{\mathrm{cycle}}
Meaning: Use H in A/m, B in tesla and volume in m³. The sign of an oriented integral depends on traversal; dissipated energy is reported as a positive magnitude.
Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.
Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.BRING THE IDEAS TOGETHER
What should stay with you
- The field and the material response must be solved consistently.
- A zero net moment can hide ordered domains whose contributions cancel.
- Ferromagnetic history cannot be represented by one constant permeability over a full cycle.
- Loop area is energy per volume per cycle; frequency and volume convert it into power.
RETRIEVE, THEN EXPLAIN
Try it without the examples
Work these out before checking. If an answer surprises you, identify which assumption or step needs another look. A correct choice is most useful when you can explain why the alternatives fail.
SELF CHECK 1
A 2.0 × 10⁻⁵ m³ specimen has net moment 0.030 A·m². Find magnetization magnitude.
Answer in A/m.
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SELF CHECK 2
A linear isotropic material has relative permeability 1.004 in SI. What is χ?
Answer in dimensionless.
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SELF CHECK 3
Which field belongs in the material law M = χH for a finite linear specimen?
- Always the applied field before the specimen is inserted
- The internal H, including the geometry-dependent response
- The flux density B in tesla substituted directly
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SELF CHECK 4
Which material observation directly identifies hysteresis in a quasistatic cycle?
- Two values of B at the same H on different sweep branches
- A nonzero current in any wire
- A positive constant susceptibility alone
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APPLY THE CHAPTER
Take the ideas into a problem
Try a complete problem next. Commit to a method before revealing a hint, then compare your reasoning with the worked solution.
The field inside a magnetized sphereAdvanced · Hints and guided reasoningRead heating from a magnetic cycleAdvanced · Hints and guided reasoningCOME BACK TO THE IDEA
What can you explain later?
Try these from memory at your next study session, then again a few days later. Say your reasoning aloud or sketch it before looking. Adjust the gap to what you can explain.
Next study session
Why can using the applied H directly in M = χH give the wrong magnetization for a finite sample?
Compare your reasoning
The material law uses internal H. The sample’s own magnetization can create a demagnetizing field, making internal H different from the applied field. Shape and constitutive response must be solved consistently.
Revisit the explanation →A few days later: change the situation
What does the area of a specified B–H hysteresis loop tell you, and what extra information converts it into heating power?
Compare your reasoning
Its area gives energy loss per unit volume per cycle in the stated quasistatic model. Multiply by sample volume and cycle frequency. Additional dynamic losses or changing loop shape need separate information.
Revisit the explanation →Compare the physical reason, the assumptions and the conclusion. Your wording can differ. If one part is missing, revisit that section and try explaining it again after a break.
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