Unit 4
Electricity and Magnetism
Class 12 Physics • Chapter 17
Magnetic Properties of Materials
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NEB/CDC syllabus scope:
Chapter 17 is a 5-teaching-hour Electricity and Magnetism chapter covering magnetic field lines and magnetic flux; magnetic flux density in a material; relative permeability and magnetic susceptibility with their relationship; hysteresis of ferromagnetic materials; and diamagnetic, paramagnetic and ferromagnetic materials.
1. Magnetic Field Lines
Magnetic field line
A magnetic field line is an imaginary curve whose tangent at any point gives the direction of the magnetic field at that point.
Properties of Magnetic Field Lines
- Outside a bar magnet, field lines are conventionally drawn from the north pole to the south pole.
- Inside the magnet, they return from south to north, forming closed loops.
- Field lines never intersect, because the field at one point has only one direction.
- Closer lines indicate a stronger magnetic field.
- Magnetic field lines are continuous; isolated magnetic poles have not been observed in ordinary magnetism.
Diagram 1: Field-line pattern of a bar magnet
2. Magnetic Flux
Magnetic flux, Φ
Magnetic flux through a surface is a measure of the magnetic field passing through that surface.
For a uniform magnetic field B through a flat area A whose normal makes angle θ with the field:
Φ = BA cosθ
SI unit:
1 weber (Wb) = 1 tesla metre² (T·m²)
Angle convention
θ is the angle between B and the normal to the surface, not between B and the surface itself.
Diagram 2: Magnetic flux through an inclined surface
3. Magnetic Flux Density in a Material
Magnetic flux density, B
Magnetic flux density describes the strength of magnetic induction in a region. Its SI unit is tesla (T).
For a uniform field perpendicular to area A:
B = Φ/A
Inside a linear magnetic material, magnetic flux density is often written as:
B = μH
where H is magnetic field intensity and μ is the permeability of the material.
| Quantity | Symbol | SI unit |
|---|---|---|
| Magnetic flux | Φ | weber (Wb) |
| Magnetic flux density | B | tesla (T) |
| Magnetic field intensity | H | A m⁻¹ |
| Permeability | μ | H m⁻¹ or N A⁻² |
| Magnetization | M | A m⁻¹ |
4. Magnetization of a Material
Magnetization, M
Magnetization is the magnetic dipole moment per unit volume of a material.
M = magnetic moment / volume
In a simple linear isotropic magnetic material:
M = χmH
where χm is magnetic susceptibility.
The magnetic flux density may also be written as:
B = μ₀(H + M)
Diagram 3: Microscopic idea of magnetization
5. Magnetic Permeability and Relative Permeability
Magnetic permeability, μ
Permeability describes how a material responds to an applied magnetic field in the relation B = μH for a linear material.
Relative permeability, μr
The ratio of the permeability of a material to the permeability of free space.
μr = μ/μ₀
Relative permeability has no unit.
| Material behavior | Typical μr relation |
|---|---|
| Diamagnetic | slightly less than 1 |
| Paramagnetic | slightly greater than 1 |
| Ferromagnetic | much greater than 1 and generally nonlinear/history-dependent |
6. Magnetic Susceptibility
Magnetic susceptibility, χm
Magnetic susceptibility is a dimensionless measure of how readily a material becomes magnetized in an applied magnetic field.
For a linear material:
χm = M/H
| Class | Sign/magnitude of χm | Response |
|---|---|---|
| Diamagnetic | small negative | weakly opposes applied field |
| Paramagnetic | small positive | weakly reinforces applied field |
| Ferromagnetic | very large positive effective response | strong magnetization; nonlinear hysteretic behavior |
7. Relation Between Relative Permeability and Susceptibility
Start with:
B = μ₀(H + M)For a linear magnetic material:
M = χmHSubstituting:
B = μ₀(H + χmH) B = μ₀(1 + χm)HBut B = μH = μ₀μrH.
Therefore:
μr = 1 + χm
or:
χm = μr − 1
Exam Important
This relation is especially useful for linear magnetic materials. Ferromagnets can be strongly nonlinear and hysteretic, so a single constant susceptibility/permeability is only an approximation over a restricted range.
Diagram 4: Derivation map for μr = 1 + χm
8. Classification of Magnetic Materials
According to their response to an applied magnetic field, common materials can be grouped as diamagnetic, paramagnetic and ferromagnetic.
| Property | Diamagnetic | Paramagnetic | Ferromagnetic |
|---|---|---|---|
| Susceptibility χm | Small negative | Small positive | Large positive effective response |
| Relative permeability μr | Slightly below 1 | Slightly above 1 | Much greater than 1 |
| Response to external field | Weakly repelled | Weakly attracted | Strongly attracted |
| Magnetization direction | Opposite H | Along H | Strongly along H via domain alignment |
| Retains magnetization? | No | Essentially no | May retain substantial magnetization |
| Examples | bismuth, copper, silver, water | aluminium, platinum, oxygen | iron, cobalt, nickel |
Diagram 5: Qualitative response of three magnetic material classes
9. Diamagnetic Materials
Diamagnetic material
A material that acquires a weak magnetization opposite to the applied magnetic field and is therefore weakly repelled from a strong-field region.
Main Characteristics
- χm is small and negative.
- μr is slightly less than 1.
- Induced magnetization opposes the external field.
- No permanent magnetic moment is required in the simple classical picture.
- Examples: bismuth, copper, silver, gold, water.
10. Paramagnetic Materials
Paramagnetic material
A material that acquires a weak magnetization in the direction of an applied magnetic field and is weakly attracted toward stronger field regions.
Main Characteristics
- χm is small and positive.
- μr is slightly greater than 1.
- Atomic or molecular magnetic moments tend to align partially with the field.
- Thermal agitation opposes complete alignment.
- Examples: aluminium, platinum and oxygen.
Curie-law idea
For many simple paramagnets over an appropriate temperature range, susceptibility decreases as temperature rises:
χm ∝ 1/T
This temperature law is useful background; the current syllabus emphasis remains the qualitative classification.
11. Ferromagnetic Materials
Ferromagnetic material
A material that can become very strongly magnetized because microscopic magnetic moments form domains that can align cooperatively.
Main Characteristics
- Very strong attraction to a magnetic field.
- Large effective positive susceptibility.
- Relative permeability may be very large.
- Magnetization is nonlinear.
- Many ferromagnets retain magnetization after the external field is removed.
- They show hysteresis.
- Examples: iron, cobalt and nickel.
Do not treat μ as constant for all ferromagnetic states
A ferromagnet is generally nonlinear and history-dependent. Its permeability can depend on field strength and previous magnetization.
12. Magnetic Domain Concept
A ferromagnetic specimen contains microscopic regions called domains. Within each domain, many magnetic moments are aligned in a preferred direction.
Unmagnetized specimen
Domains may point in different directions, so their vector sum can be small.
Magnetized specimen
An external magnetic field favors domains aligned with the field. Domain-wall motion and rotation increase the net magnetization.
Diagram 6: Domain arrangement before and after magnetization
13. Hysteresis of Ferromagnetism
Magnetic hysteresis
Hysteresis is the lag of magnetic flux density B or magnetization M behind the magnetizing field H when a ferromagnetic material is taken through a cycle of magnetization.
Because the magnetic state depends on the material’s previous history, increasing and decreasing H do not follow the same B–H path.
Important Terms
| Term | Meaning |
|---|---|
| Saturation | Region where increasing H produces comparatively little further increase in magnetization |
| Retentivity / remanence | Residual B or M remaining when H is reduced to zero after strong magnetization |
| Coercivity | Magnitude of reverse H required to reduce residual magnetization/flux density to zero |
| Hysteresis loss | Energy dissipated per cycle per unit volume; related to the area enclosed by the B–H loop |
Core exam point
Area of the B–H hysteresis loop represents energy loss per unit volume per magnetization cycle.
14. B–H Hysteresis Loop
Diagram 7: Saturation, remanence and coercivity on a hysteresis loop
Sequence through a Magnetization Cycle
- Increase H from an unmagnetized state: B rises toward positive saturation.
- Reduce H to zero: a residual value Br remains.
- Apply reverse H: B falls to zero at coercive field −Hc.
- Increase reverse H further: negative saturation is approached.
- Reverse the field again: a complete closed hysteresis loop is formed.
15. Soft and Hard Magnetic Materials
| Property | Soft magnetic material | Hard magnetic material |
|---|---|---|
| Hysteresis loop | Narrow | Wide |
| Coercivity | Low | High |
| Hysteresis loss | Low | Higher |
| Magnetization reversal | Easy | Difficult |
| Typical use | Transformer/inductor/electromagnet cores | Permanent magnets |
Diagram 8: Narrow and wide hysteresis loops
16. Practical Applications
Transformer cores
Use soft magnetic materials with low hysteresis loss and high permeability.
Electromagnets
Soft magnetic cores become strongly magnetized and demagnetize readily.
Permanent magnets
Hard magnetic materials require high retentivity and coercivity.
Magnetic shielding / field guiding
High-permeability materials can provide a preferred magnetic-flux path in appropriate designs.
17. Worked Examples and Numericals
Example 1: Magnetic Flux
A uniform magnetic field B = 0.50 T passes through an area A = 0.020 m². The field makes 60° with the surface normal.
Φ = BA cosθ Φ = 0.50 × 0.020 × cos60° Φ = 5.0 × 10⁻³ Wb
Example 2: Relative Permeability
A material has μ = 5.0 × 10⁻⁶ H m⁻¹. Take μ₀ = 4π × 10⁻⁷ H m⁻¹.
μr = μ/μ₀ μr = (5.0 × 10⁻⁶)/(4π × 10⁻⁷) μr ≈ 3.98
Example 3: Susceptibility from Relative Permeability
If μr = 1.00035:
χm = μr − 1 = 3.5 × 10⁻⁴The small positive susceptibility indicates paramagnetic behavior.
Example 4: Magnetization
A linear magnetic material has χm = 2.0 × 10⁻³ and H = 4.0 × 10⁴ A m⁻¹.
M = χmH M = 2.0 × 10⁻³ × 4.0 × 10⁴ M = 80 A m⁻¹
Example 5: Identify the Material
A sample has χm = −1.2 × 10⁻⁵.
Since susceptibility is small and negative, the material is diamagnetic.
Example 6: Hysteresis Choice
Which material should be chosen for a transformer core: one with a narrow loop or one with a wide loop?
Answer: a narrow-loop soft magnetic material, because hysteresis energy loss per cycle is smaller.
18. Complete Formula Sheet
| Topic | Formula |
|---|---|
| Magnetic flux | Φ = BA cosθ |
| Flux density for perpendicular uniform flux | B = Φ/A |
| Linear magnetic medium | B = μH |
| Magnetization definition | M = magnetic moment / volume |
| Magnetization in linear medium | M = χmH |
| Field in magnetic material | B = μ₀(H + M) |
| Relative permeability | μr = μ/μ₀ |
| Susceptibility | χm = M/H |
| Permeability–susceptibility relation | μr = 1 + χm |
19. Common Exam Mistakes
- Confusing magnetic flux Φ with magnetic flux density B.
- Using Φ = BA sinθ when θ is defined from the surface normal. Correct: Φ = BA cosθ.
- Writing the SI unit of magnetic flux as tesla. Flux is measured in weber; flux density in tesla.
- Confusing B and H. B is flux density; H is magnetizing field intensity.
- Writing relative permeability with units. μr is dimensionless.
- Writing susceptibility with units. χm is dimensionless.
- Using μr = χm instead of μr = 1 + χm.
- Confusing diamagnetism and paramagnetism: diamagnetic χm is negative, paramagnetic χm is positive.
- Calling a diamagnetic material strongly repelled. The effect is usually weak.
- Saying an N-pole field line ends permanently at S. Magnetic field lines form closed loops.
- Claiming field lines intersect. They do not.
- Confusing ferromagnetic domain alignment with ordinary paramagnetic alignment.
- Writing that a ferromagnet always has a single constant μ and χ. Ferromagnetic response is generally nonlinear and hysteretic.
- Confusing retentivity with coercivity.
- Writing that coercivity is residual magnetization. Coercivity is the reverse field needed to reduce B/M to zero.
- Forgetting that the hysteresis-loop area represents energy loss per unit volume per cycle.
- Choosing a wide hysteresis loop for a transformer core. Transformer cores require low hysteresis loss.
- Choosing a narrow loop for a strong permanent magnet without considering retentivity and coercivity.
20. Important Exam Questions
Very Short / Short Questions
- Define a magnetic field line.
- State four properties of magnetic field lines.
- Define magnetic flux and state its SI unit.
- Write Φ = BA cosθ and explain each symbol.
- Define magnetic flux density.
- State the SI unit of B.
- Define magnetic field intensity H.
- Define magnetization M.
- Define magnetic permeability.
- Define relative permeability.
- Define magnetic susceptibility.
- Write the relation between M, H and χm.
- Derive μr = 1 + χm.
- What is a diamagnetic material?
- What is a paramagnetic material?
- What is a ferromagnetic material?
- Give two examples each of dia-, para- and ferromagnetic materials.
- What is a magnetic domain?
- Define hysteresis.
- Define retentivity.
- Define coercivity.
- What does the area of a hysteresis loop represent?
- Differentiate soft and hard magnetic materials.
- Why are soft magnetic materials used in transformer cores?
- Why are hard magnetic materials used for permanent magnets?
Long Questions / Derivations
- Explain magnetic field lines and magnetic flux with diagrams.
- Explain magnetic flux density in a magnetic material.
- Define relative permeability and susceptibility and derive their relationship.
- Compare diamagnetic, paramagnetic and ferromagnetic materials in tabular form.
- Explain ferromagnetism using the domain concept.
- Explain magnetic hysteresis with a labelled B–H loop.
- Define saturation, retentivity and coercivity using a hysteresis loop.
- Explain the significance of hysteresis-loop area.
- Compare soft and hard magnetic materials and state their uses.
Numerical Practice
- Calculate magnetic flux from B, A and θ.
- Calculate B from magnetic flux and area.
- Calculate relative permeability from μ and μ₀.
- Calculate susceptibility from relative permeability.
- Calculate magnetization from χm and H.
- Identify material class from the sign/magnitude of susceptibility.
Exam Strategy
Focus on four anchors: Φ = BAcosθ → μr and χm → dia/para/ferro comparison → hysteresis loop. The most valuable diagram is the labelled B–H loop showing saturation, remanence, coercivity and loop area.
21. One-Minute Revision
- Unit 4, Chapter 17: Magnetic Properties of Materials.
- Magnetic field lines form continuous closed loops.
- Magnetic flux: Φ = BA cosθ.
- Flux unit = weber (Wb).
- Flux density B unit = tesla (T).
- In a linear material: B = μH.
- Magnetization M = magnetic moment per unit volume.
- For a linear material: M = χmH.
- B = μ₀(H + M).
- Relative permeability μr = μ/μ₀.
- Magnetic susceptibility χm = M/H.
- Core relation: μr = 1 + χm.
- Diamagnetic: χm small negative, μr slightly less than 1.
- Paramagnetic: χm small positive, μr slightly greater than 1.
- Ferromagnetic: very strong positive response and domain alignment.
- Common diamagnetic examples: bismuth, copper, water.
- Common paramagnetic examples: aluminium, platinum, oxygen.
- Common ferromagnetic examples: iron, cobalt, nickel.
- Ferromagnets contain magnetic domains.
- Hysteresis is the lag of B/M behind H.
- Retentivity = residual magnetization/flux density at H = 0.
- Coercivity = reverse field needed to reduce B/M to zero.
- Loop area represents energy loss per unit volume per cycle.
- Soft magnetic materials have narrow loops and low coercivity.
- Hard magnetic materials have wide loops and high coercivity.
- Transformer cores need low hysteresis loss.
- Permanent magnets need strong retention and coercivity.
22. Diagram Practice
Students should practice these labelled diagrams for the NEB examination:
- Magnetic field lines around a bar magnet.
- Magnetic flux through an inclined area.
- Dipole alignment and magnetization.
- Relation between B, H, M, χm and μr.
- Diamagnetic, paramagnetic and ferromagnetic responses.
- Ferromagnetic domain alignment.
- Labelled B–H hysteresis loop.
- Soft and hard magnetic hysteresis loops.
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