Magnetism and Matter: NEET notes

1. Magnets and their poles

NCERT §5.1

• Magnetic fields are everywhere, from galaxies to atoms. The word magnet comes from Magnesia, a Greek island where magnetic ore was found as early as 600 BC.
• The earth acts as a magnet whose field points roughly from the geographic south to the geographic north.
• A freely hung bar magnet settles along the north–south line. The end that turns towards geographic north is its north pole; the other end is its south pole.
• Like poles (N and N, or S and S) repel each other; unlike poles attract.
• A pole cannot be isolated. Break a bar magnet in two and each half is a complete, somewhat weaker magnet with its own N and S. Isolated poles, called magnetic monopoles, are not known to exist, unlike separate positive and negative charges.
• Iron and its alloys can be made into magnets.
• Example 5.1: cutting a magnet across its length or along it gives two magnets either way, each with a north and a south pole.

2. Magnetic field lines

NCERT §5.2, §5.2.1

• Iron filings sprinkled on glass over a short bar magnet line up in a pattern much like the field of an electric dipole: the magnet has two poles, as the dipole has two charges. A current-carrying solenoid gives a very similar pattern.
• The pattern traces the magnetic field lines, a picture of B. They form continuous closed loops, running outside from N to S and inside the magnet from S back to N. Electric field lines of a dipole instead begin on the positive charge and end on the negative one or go off to infinity.
• The tangent to a field line at any point gives the direction of the net field B there.
• The more lines cross a unit area, the stronger B is: the lines crowd near the poles.
• Two field lines never cross, since B would then have two directions at one point.
• Field lines can be plotted with a small compass: the needle sets itself along the line at each spot.
• They are not lines of force: the force on a moving charge, qv × B, is perpendicular to B, not along it (Example 5.4).

3. Bar magnet as an equivalent solenoid

NCERT §5.2.2

• A current loop is a magnetic dipole, and Ampere suggested that all magnetism comes from circulating currents. The field lines of a bar magnet and of a finite solenoid look alike, so a bar magnet can be pictured as a great many circulating currents, like the turns of a solenoid.
• Cutting a magnet in half is like cutting a solenoid in half: two shorter solenoids, each weaker, with the lines still leaving one face and entering the other.
• A small compass moved round a bar magnet and round a finite solenoid swings in the same way in both cases.
• Far from a finite solenoid, at distance r along its axis, the field is B = (μ₀/4π)(2m/r³), where m is the solenoid's magnetic moment (NIA for N turns of area A carrying I). A bar magnet's far axial field, measured in the lab, has the same form.
• So a bar magnet's magnetic moment equals that of the solenoid which would make the same field; its unit is A m², also written J/T.

4. Dipole in a uniform field

NCERT §5.2.3

• A compass needle of moment m in a uniform field B feels no net force but a torque τ = m × B, of size mB sin θ, where θ is the angle between m and B. It is a restoring torque: it turns m towards B.
• The magnetic potential energy follows by integrating the torque over angle: U = −mB cos θ = −m·B. The zero is chosen at θ = 90°, where the needle is at right angles to the field.
• U is least, −mB, at θ = 0°: the most stable position. It is greatest, +mB, at θ = 180°: the most unstable position. Turning the magnet from 0° to 180° takes work 2mB.
• The torque is zero at both 0° and 180°, but only 0° is stable: a small nudge from 180° grows.
• Set oscillating, the needle swings about the field direction; the arrangement can be used to find B if m is known, or m if B is known.
• Example 5.1: in a uniform field a magnetised needle is only turned, never pulled. An iron nail near a bar magnet is in a non-uniform field; the magnet induces a moment in it, and because the nail's induced pole nearest the magnet is the unlike one, the nail is drawn in as well as turned.
• Example 5.1: a field source need not have a north and a south pole; that happens only if it has a net magnetic moment, which a toroid or an infinite straight wire does not.
• Example 5.1: of two identical-looking bars, a repulsion at some pairing shows both are magnets. If only one is, touch the end of one bar to the middle of the other: a magnet's field is strongest at its ends and weakest at its centre, so no pull at the middle of B means B is the magnet.

5. The electrostatic analog

NCERT §5.2.4

• The far field of a bar magnet of moment m follows from the electric dipole's field by swapping E → B, p → m and 1/(4πε₀) → μ₀/4π.
• On the axis, at distance r much larger than the magnet's size l: B_A = (μ₀/4π)(2m/r³), along m.
• On the equatorial line (the normal bisector), for r ≫ l: B_E = −(μ₀/4π)(m/r³), half the axial size at the same distance and pointing opposite to m.
• Both fall off as 1/r³: doubling the distance cuts the field to one-eighth.
• The dipole analogy runs through the table: 1/ε₀ ↔ μ₀, p ↔ m, torque p × E ↔ m × B, energy −p·E ↔ −m·B.
• Example 5.2: a needle Q near a needle P is in equilibrium only when m_Q is parallel or antiparallel to P's field at Q. Parallel is stable, antiparallel unstable, and the lowest energy comes with Q on P's axis, pointing along P's field there, where that field is twice its equatorial size.

6. Gauss's law for magnetism

NCERT §5.3

• For an electric dipole, a closed surface round the positive charge has net outward flux, q/ε₀, while a surface enclosing no net charge has as many lines entering as leaving.
• Magnetic field lines are continuous closed loops, so any closed surface, even one round a single pole of a magnet or an end of a solenoid, has as many lines entering as leaving.
• The flux through a small area ΔS is ΔφB = B·ΔS. Gauss's law for magnetism: the total magnetic flux through any closed surface is zero, ΣB·ΔS = 0.
• Why: no isolated magnetic pole (monopole) has ever been found, so B has neither sources nor sinks. The most basic magnetic unit is a current loop or dipole, and every magnetic effect can be built from arrangements of these.
• Example 5.3: magnetic lines cannot spread out from a point or from one plate, cannot cross, and a static field line can form a closed loop only round a region carrying current. Lines inside a toroid are correct; lines of a solenoid or between pole pieces that stay perfectly straight at the ends break Ampere's law, so the ends must fringe.
• Example 5.4: if monopoles existed, the closed-surface flux would equal μ₀q_m, where q_m is the magnetic charge enclosed.
• Example 5.4: no element of a wire feels a force from its own field, though it can feel one from other elements of the same wire (zero for a straight wire). A system with zero net charge can still have a magnetic moment, as paramagnetic atoms do.

7. Magnetisation and magnetic intensity

NCERT §5.4

• Orbiting electrons give atoms magnetic moments, and in bulk matter these add as vectors. The magnetisation M of a sample is its net magnetic moment per unit volume, M = m_net/V: a vector with dimensions L⁻¹ A, measured in A m⁻¹.
• In an empty long solenoid B₀ = μ₀nI. Filling it with a magnetised material adds a field B_m = μ₀M, so B = B₀ + B_m.
• The magnetic intensity is defined as H = B/μ₀ − M, also in A m⁻¹. Then B = μ₀(H + M): H stands for the external cause (in a solenoid H = nI, whatever the core), M for the material's response.
• For most materials M is proportional to H: M = χH, where the magnetic susceptibility χ is dimensionless. χ is small and positive for paramagnetic materials, small and negative for diamagnetic ones (M opposite to H).
• So B = μ₀(1 + χ)H = μ₀μᵣH = μH, with relative permeability μᵣ = 1 + χ (the magnetic twin of the dielectric constant) and permeability μ = μ₀μᵣ, in the same units as μ₀.
• χ, μᵣ and μ are linked, so knowing any one gives the other two.
• Example 5.5: a core of μᵣ = 400 in windings of 1000 turns per metre carrying 2 A gives H = nI = 2 × 10³ A/m, B = μᵣμ₀H = 1.0 T, and M = (μᵣ − 1)H = 399H ≈ 8 × 10⁵ A/m.
• Example 5.5: the magnetising current is the extra current that would give the same B with no core, from B = μ₀n(I + I_M); it comes to I_M = 794 A, showing how much the core adds.

8. Diamagnetism

NCERT §5.5, §5.5.1

• By susceptibility: diamagnetic if χ is negative (−1 ≤ χ < 0, 0 ≤ μᵣ < 1, μ < μ₀), paramagnetic if χ is small and positive (μᵣ just above 1, μ > μ₀), ferromagnetic if χ ≫ 1 (μᵣ ≫ 1, μ ≫ μ₀).
• A diamagnetic substance tends to move from the stronger to the weaker part of a field: a magnet repels it. In a field, the lines are pushed out of it and the field inside is reduced, usually by about one part in 10⁵.
• Explanation: in a diamagnetic atom the orbital moments of the electrons cancel. An applied field slows the electrons whose moments lie along it and speeds up those opposing it (an induced effect, following Lenz's law), leaving a net moment opposite to the field, hence the repulsion.
• Examples: water, sodium chloride, nitrogen at STP, and the solids bismuth, lead, copper and silicon.
• Diamagnetism is present in every substance, but it is so weak that para- or ferromagnetism usually hides it.
• Superconductors, metals cooled to very low temperatures, are perfect conductors and perfect diamagnets: the field is expelled completely, χ = −1 and μᵣ = 0. This is the Meissner effect. A superconductor repels a magnet and is repelled by it; superconducting magnets can be used for magnetically levitated fast trains.
• The two opposite behaviours come from a tiny difference in χ: about −10⁻⁵ for diamagnetic materials against about +10⁻⁵ for paramagnetic ones.

9. Paramagnetism

NCERT §5.5.2

• Paramagnetic substances are weakly magnetised in a field and tend to move from a weak-field region to a strong-field one: a magnet attracts them weakly.
• Each atom, ion or molecule carries a permanent magnetic dipole moment of its own, but ceaseless thermal motion keeps them randomly pointed, so there is no net magnetisation without a field.
• A strong enough applied field B₀, especially at low temperature, lines the atomic moments up along B₀. The lines crowd slightly into the sample and the field inside rises, usually by about one part in 10⁵.
• In a non-uniform field a paramagnetic bar moves from weak field to strong.
• Examples: aluminium, sodium, calcium, oxygen (at STP) and copper chloride.
• χ and μᵣ of a paramagnet depend on the material and, in a simple way, on its temperature. A stronger field or a lower temperature raises M until it saturates, with every dipole aligned.

10. Ferromagnetism

NCERT §5.5.3

• Ferromagnetic substances are strongly magnetised in a field and are strongly drawn from weak-field to strong-field regions.
• Their atoms carry dipole moments, as in a paramagnet, but the moments interact and line up spontaneously over a macroscopic region called a domain (the explanation needs quantum mechanics). A typical domain is about 1 mm across and holds about 10¹¹ atoms.
• At first the domains point randomly and there is no net magnetisation. An applied field B₀ turns the domains towards it, and those already along B₀ grow, until they merge into one giant domain. Domains and their motion can be seen under a microscope using a liquid suspension of ferromagnetic powder.
• The field lines crowd strongly into a ferromagnet, and in a non-uniform field it moves towards the high-field region.
• Hard ferromagnets stay magnetised after the field is switched off, so they make permanent magnets such as compass needles. Examples: lodestone, found in nature, and alnico, whose ingredients are aluminium, nickel, cobalt, copper and iron.
• Soft ferromagnets, such as soft iron, lose their magnetisation when the field is removed.
• Ferromagnetic elements include iron, cobalt, nickel and gadolinium, with relative permeability above 1000.
• Heated enough, a ferromagnet becomes a paramagnet as its domain structure breaks up; the magnetisation fades gradually with temperature.
• Substances that keep their ferromagnetism at room temperature for a long time are permanent magnets.

Must-know facts

1. Magnetic monopoles are not known to exist; a broken magnet gives two complete magnets.
2. Field lines are closed loops, N to S outside and S to N inside; they never cross.
3. A bar magnet behaves like a solenoid of the same moment m = NIA.
4. In a uniform field a magnet feels torque mB sin θ but zero net force.
5. U = −mB cos θ, zero at 90°; −mB at 0° (stable), +mB at 180° (unstable).
6. Work to turn a magnet from along B to opposite B is 2mB.
7. Far axial field (μ₀/4π)(2m/r³); equatorial (μ₀/4π)(m/r³), opposite to m.
8. Replace E → B, p → m, 1/4πε₀ → μ₀/4π to get magnetic dipole results.
9. Gauss's law for magnetism: net flux of B through any closed surface is zero.
10. M = m_net/V, unit A m⁻¹; H = B/μ₀ − M, unit A m⁻¹; B = μ₀(H + M).
11. M = χH; μᵣ = 1 + χ; μ = μ₀μᵣ.
12. In a solenoid H = nI whatever the core.
13. Diamagnetic: χ negative, μᵣ < 1, repelled; bismuth, copper, lead, water, NaCl, N₂.
14. Paramagnetic: χ small positive, weakly attracted; Al, Na, Ca, O₂, CuCl₂; χ depends on temperature.
15. Ferromagnetic: χ ≫ 1, μᵣ above 1000; Fe, Co, Ni, Gd; domains of about 1 mm, 10¹¹ atoms.
16. Superconductor: χ = −1, μᵣ = 0, field fully expelled (Meissner effect).
17. Hard ferromagnets (alnico, lodestone) make permanent magnets; soft iron loses its magnetism.
18. A ferromagnet heated enough becomes paramagnetic.

Common traps

Formulas

Key terms

Magnetic monopole

An isolated north or south pole; none is known to exist.

Magnetic field line

A closed curve whose tangent at each point gives the direction of B; closer lines mean a stronger field.

Magnetic moment (m)

The vector that fixes a magnet's torque and far field; NIA for an equivalent solenoid, unit A m² or J/T.

Magnetic potential energy

U = −m·B for a dipole in a uniform field, taken as zero when m is at right angles to B.

Equatorial line

The normal bisector of a magnet, where its far field is half the axial value and opposite to m.

Magnetic flux

B·ΔS summed over a surface; unit weber (T m²).

Gauss's law for magnetism

The net magnetic flux out of any closed surface is zero.

Magnetisation (M)

Net magnetic moment per unit volume of a sample, in A m⁻¹.

Magnetic intensity (H)

B/μ₀ − M, the part of the field set by external currents; nI in a solenoid.

Magnetic susceptibility (χ)

The dimensionless ratio M/H, measuring how strongly a material responds to a field.

Relative permeability (μᵣ)

1 + χ, the factor by which a material multiplies μ₀; the magnetic twin of the dielectric constant.

Diamagnetic

Having small negative χ; pushed from strong field to weak.

Paramagnetic

Having small positive χ from permanent atomic moments; weakly pulled into a strong field.

Ferromagnetic

Having χ ≫ 1 because atomic moments align in domains; strongly attracted.

Domain

A region, typically about 1 mm across with about 10¹¹ atoms, in which all atomic moments of a ferromagnet point one way.

Meissner effect

The complete expulsion of a magnetic field from a superconductor, which is a perfect diamagnet.

Hard and soft ferromagnets

Hard ones keep their magnetisation when the field is removed (alnico, lodestone); soft ones lose it (soft iron).