Magnetism and Matter

Magnetism was known long before electricity. The word comes from Magnesia, a district in Greece where the mineral magnetite (lodestone, Fe₃O₄) was found; the Chinese used lodestone compasses for navigation more than two thousand years ago. NCERT opens the chapter with a handful of facts that every later section builds on, and the examiner assumes you know them.

First, the earth behaves as a magnet. Its field lines point, roughly, from the geographic south to the geographic north, so the end of a freely suspended magnet that points towards geographic north is called the north pole of the magnet. Second, a bar magnet has two poles, north and south; like poles repel and unlike poles attract. Third, and this is the fact behind Gauss's law for magnetism, magnetic poles never occur singly. If you cut a magnet into two you do not get an isolated north and an isolated south; you get two smaller magnets, each with both poles. Isolated magnetic charges (monopoles) have never been observed. Fourth, you can make a magnet from an iron bar by keeping it in a magnetic field or by stroking it with a magnet, and some materials keep the magnetism once the field is removed.

The chapter then does three things. It treats the bar magnet as a magnetic dipole, exactly parallel to the electric dipole of Chapter 1, and shows that a current-carrying solenoid is the same kind of object. It states Gauss's law for magnetism. And it asks how a piece of matter responds when placed in a magnetic field, which leads to magnetisation, susceptibility, permeability and the three classes of magnetic materials. Note that in the rationalised NCERT text the sections on the earth's magnetism (declination, dip), hysteresis, and permanent magnets and electromagnets have been dropped, so the board paper does not ask them. Older question banks still contain them; do not waste revision time there.

Sprinkle iron filings on a sheet placed over a bar magnet and they arrange themselves in curved lines that bunch near the two ends. Those ends are the poles. The same pattern is produced by a current-carrying solenoid, which is the first hint that a bar magnet and a solenoid are the same physical object seen from outside.

NCERT describes a bar magnet by a magnetic moment m, a vector directed from the south pole to the north pole inside the magnet. If the pole strength is q_m and the magnetic length is 2l (the distance between the poles, a little less than the geometric length), then m = q_m × 2l. The SI unit of m is A m², which is the same as J T⁻¹. Pole strength has the unit A m. You do not need to derive anything with q_m; it matters only for the cut-the-magnet questions.

Two facts about cutting and bending a magnet are asked again and again. Cut a magnet transverse to its length into two equal halves: each piece keeps the same pole strength q_m, its length becomes l, so its moment becomes m/2. Cut it along its length: each half has half the cross-section, so its pole strength becomes q_m/2, its length stays 2l, and its moment is again m/2. Bend a magnet of length L and moment m into a semicircle: the pole strength is unchanged, but the poles are now a chord 2R = 2L/π apart, so the new moment is 2m/π. A magnetic needle placed near a magnet aligns along the field of the magnet; this is how the field pattern is mapped point by point.

Magnetic field lines are the pictorial way of showing the field B, and NCERT lists their properties in four points that you should be able to write from memory.

1. The magnetic field lines of a magnet (or a solenoid) form continuous closed loops. Outside the magnet they run from the north pole to the south pole; inside the magnet they continue from the south pole back to the north pole. This is the key difference from electric field lines, which begin on positive charges and end on negative charges and are never closed loops.
2. The tangent to a field line at any point gives the direction of the net magnetic field B at that point.
3. The larger the number of field lines crossing per unit area, the larger the magnitude of B. The lines are crowded near the poles, where the field is strongest.
4. Field lines never intersect. If they did, the field would have two directions at the crossing point, which is impossible.

The field-line pictures for a bar magnet and for a finite solenoid look identical from outside, and the field of an electric dipole has the same shape outside the dipole. Only inside do they differ: inside the magnet the lines run S to N to close the loop, while between the charges of an electric dipole the field runs from positive to negative, opposite to the dipole moment direction. NCERT points out that field lines are not really lines of force; a compass needle placed on a line experiences a torque, not a force along the line.

When the exam asks you to draw the field lines of a bar magnet, draw the loops passing through the magnet, mark N and S, put arrows N to S outside and S to N inside, and make the lines closer together near the poles. A sketch without the inside lines loses a mark because it fails to show the closed-loop property.

Ampere's hypothesis is that all magnetic phenomena come from circulating currents. The evidence in this chapter is that the field of a bar magnet, at points far away, has exactly the form of the field of a solenoid, and both are dipole fields. The rationalised syllabus asks only for a qualitative treatment, but the derivation is short and has been asked in the past, so learn the steps.

Take a solenoid of radius a, length 2l, n turns per unit length and current I, with its centre at the origin and its axis along x. Consider a thin slice of the solenoid of thickness dx at a distance x from the centre; it contains n dx turns and acts as a circular coil. From Chapter 4, the field of this slice at a point P on the axis at distance r from the centre is dB = μ₀ (n dx) I a² / 2[(r − x)² + a²]^3/2. For a point far from the solenoid, r ≫ a and r ≫ l, so the denominator becomes simply 2r³ and dB ≈ μ₀ n I a² dx / 2r³. Integrating x from −l to +l gives the total field on the axis:

B = μ₀ n I a² (2l) / 2r³ = (μ₀/4π) · 2 [n (2l) I (πa²)] / r³ = (μ₀/4π) · 2m/r³,

where m = n(2l) I (πa²) = N I A is the magnetic moment of the solenoid (N total turns, area A). This is exactly the axial field of a bar magnet of moment m. So a solenoid behaves like a bar magnet with moment NIA, directed along its axis in the sense given by the right-hand rule (curl the fingers along the current, the thumb points to the north face). NCERT confirms the equivalence experimentally: a suspended solenoid settles N-S like a magnet, and it attracts or repels another magnet just as a bar magnet does.

Two exam uses follow. A short-answer question gives N, I and A and asks for m and then the torque in a field. A conceptual question asks how a solenoid is like a bar magnet: same field pattern outside, both have a north and a south face, both experience the same torque in a field, and the solenoid's poles reverse when the current reverses.

Place a small compass needle of moment m and moment of inertia I in a uniform field B, with m making an angle θ with B. Think of the needle as two poles ±q_m separated by 2l: the field pushes the north pole along B and the south pole opposite to B with equal forces q_mB. The net force is zero, so the needle does not move as a whole, but the two forces form a couple of arm 2l sin θ. The torque is τ = q_mB × 2l sin θ = mB sin θ, and as a vector τ = m × B. The torque tends to align m with B. It is zero at θ = 0 (stable equilibrium) and θ = 180° (unstable equilibrium), and maximum, τ = mB, at θ = 90°.

Oscillations. Displace the needle by a small angle θ and release it. The restoring torque gives I d²θ/dt² = −mB sin θ. For small θ, sin θ ≈ θ, so d²θ/dt² = −(mB/I) θ. This is the SHM equation with ω² = mB/I, so the time period is T = 2π√(I/mB). Rearranging, B = 4π²I/(mT²), which is how a vibration magnetometer measures a field. This full argument, from the couple to the period, is the standard 5-mark derivation from this chapter.

Potential energy. The work done by an external agent in turning the dipole from θ₁ to θ₂ is W = ∫ mB sin θ dθ = mB (cos θ₁ − cos θ₂). Taking the zero of energy at θ = 90° (where the sin term integrates to zero) gives U_m = −mB cos θ = −m·B. U is minimum, −mB, at θ = 0, the most stable position; it is maximum, +mB, at θ = 180°, the most unstable position; and it is zero at θ = 90°. The work required to turn the magnet from the stable to the unstable position is 2mB.

Numericals from this section are the commonest in the chapter. Read the angle carefully: the angle given is between m and B; if the question says the axis is perpendicular to the field, sin θ = 1. And keep the sign of U: a negative answer is correct at the stable position.

Compare the results for the bar magnet with those for the electric dipole in Chapter 1 and you will see that every magnetic formula is obtained from the electric one by three replacements: E → B, p → m and 1/ε₀ → μ₀. NCERT summarises this in Table 5.1, and reproducing that table is a safe 3-mark answer.

• Equatorial field of a short dipole at distance r: electric E = −p/(4πε₀r³); magnetic B_E = −(μ₀/4π) · m/r³. The minus sign says the field on the equatorial line is antiparallel to the moment.
• Axial field: electric E = 2p/(4πε₀r³); magnetic B_A = (μ₀/4π) · 2m/r³, parallel to m.
• Torque in an external field: p × E and m × B.
• Potential energy: −p·E and −m·B.

Both fields fall as 1/r³, and at the same distance the axial field is twice the equatorial field. The exact expressions for a magnet of half-length l are B_A = (μ₀/4π) · 2mr/(r² − l²)² and B_E = (μ₀/4π) · m/(r² + l²)^3/2; for r ≫ l these reduce to the forms above.

NCERT works Example 5.4 with this: a bar magnet of length 5.0 cm and moment 0.40 A m², at 50 cm from its mid-point. Since l = 2.5 cm ≪ 50 cm, B_E = 10⁻⁷ × 0.40/(0.5)³ = 3.2 × 10⁻⁷ T and B_A = 6.4 × 10⁻⁷ T. The analogy also tells you what is not the same: an electric dipole is made of two real charges that can be separated, while the magnetic dipole cannot be split into monopoles, and inside the magnet the field runs S to N, along m, whereas inside an electric dipole the field runs opposite to p.

In electrostatics, Gauss's law says that the flux of E through a closed surface equals the enclosed charge divided by ε₀. The magnetic version is simpler: the net magnetic flux through any closed surface is zero. In symbols, Φ_B = ∮ B·ΔS = 0, where the sum runs over all the area elements of the closed surface.

The reason is in the field-line picture. Because magnetic field lines are closed loops, every line that enters a closed surface must also leave it. The number of lines going in equals the number coming out, so the net flux is zero, whatever surface you draw and wherever you draw it. Try it on a bar magnet: a closed surface around only the north pole has lines leaving through the outer part and the same lines entering through the part that cuts the magnet. Compare an electric dipole: a surface enclosing only the positive charge has net outward flux q/ε₀, because the lines start on that charge.

The physical content of the law is that isolated magnetic poles, magnetic monopoles, do not exist. The simplest source of a magnetic field is a dipole (a current loop), not a magnetic charge. This is why the law has a zero on the right-hand side and why there is no magnetic equivalent of a point charge.

Two things are asked in the exam. First, state the law and give its meaning in one line. Second, a figure-based question: given a set of field-line diagrams, say which cannot represent a magnetic field. Any diagram in which lines start or end at a point in empty space, or in which the net flux out of a closed region is not zero, cannot be a magnetic field; it could be an electrostatic field with charges at those points. Diagrams with intersecting lines cannot represent either field.

Every electron in an atom has an orbital magnetic moment and a spin magnetic moment, so each atom may have a net moment. In most materials these atomic moments point in random directions and cancel; in some they add up to a bulk moment. The magnetisation M of a sample is its net magnetic moment per unit volume, M = m_net/V, a vector with SI unit A m⁻¹.

Now put a material inside a long solenoid of n turns per metre carrying current I. The field produced by the current alone is B₀ = μ₀nI. The material becomes magnetised and adds its own field B_m = μ₀M (for a long, uniformly magnetised sample), so the total field inside is B = B₀ + B_m = μ₀(nI + M). It is convenient to define the magnetic intensity H = B/μ₀ − M. H depends only on the free current in the windings, H = nI for the solenoid, and it has the same unit as M, A m⁻¹. The total field is then B = μ₀(H + M): H is the external contribution and M is the material's own contribution. Note the pattern, sometimes written as B = μ₀(H + M) with M itself depending on H.

For most materials M is proportional to H: M = χH, where χ is the magnetic susceptibility, a dimensionless number that measures how easily the material is magnetised. Then B = μ₀(1 + χ)H = μ₀μ_rH = μH, with μ_r = 1 + χ the relative permeability (dimensionless) and μ = μ₀μ_r the permeability of the material (T m A⁻¹). The three quantities χ, μ_r and μ are not independent; give any one and the other two follow.

NCERT's worked example (Example 5.9 in the older text) is the template for the 3-mark numerical. A solenoid has 1000 turns per metre, current 2 A, and a core of μ_r = 400. Then H = nI = 2000 A m⁻¹; B = μ_rμ₀H = 400 × 4π × 10⁻⁷ × 2000 ≈ 1.0 T; M = (μ_r − 1)H ≈ 8 × 10⁵ A m⁻¹. The magnetising current I_m, the extra current in the windings that would produce the same B without the core, satisfies M = nI_m, so I_m = 7.98 × 10⁵/1000 ≈ 798 A (about 800 A). That number is the whole point: the core does the work of hundreds of amperes.

Materials are classified by their susceptibility χ. NCERT gives the ranges in one line each, and the same line is what the examiner wants when a question gives a value of χ or μ_r and asks you to identify the material.

• Diamagnetic: −1 ≤ χ < 0, so 0 ≤ μ_r < 1 and μ < μ₀. Small negative susceptibility; feebly repelled by a magnet.
• Paramagnetic: 0 < χ < ε, so 1 < μ_r < 1 + ε and μ > μ₀, where ε is a small positive number. Small positive susceptibility; feebly attracted.
• Ferromagnetic: χ ≫ 1, so μ_r ≫ 1 and μ ≫ μ₀. Large positive susceptibility (hundreds to thousands); strongly attracted.

So a material with μ_r = 0.9983 (χ = −0.0017) is diamagnetic, one with χ = 1.9 × 10⁻⁵ is paramagnetic, and one with μ_r = 400 is ferromagnetic. The rule is mechanical: negative means dia, small positive means para, large positive means ferro.

Three more distinctions carry the 2- and 3-mark questions. Field lines: in a diamagnetic sample the lines are pushed out and become sparser inside (the field inside is reduced); in a paramagnetic sample they are drawn in slightly and become denser inside; in a ferromagnetic sample they are pulled in strongly and crowd inside. Non-uniform field: a diamagnetic sample moves from the stronger to the weaker part of the field, while para- and ferromagnetic samples move from the weaker to the stronger part. A diamagnetic rod suspended freely sets itself perpendicular to the field; a paramagnetic rod sets itself along the field. Temperature: diamagnetism is almost independent of temperature; paramagnetic susceptibility falls as 1/T (Curie's law); a ferromagnet loses its ferromagnetism above the Curie temperature and becomes paramagnetic.

Learn a few examples from each class with their signs: Bi, Cu, Pb, Si, water, nitrogen (at STP), NaCl and diamond are diamagnetic; Al, Na, Ca, O₂ (at STP), CuCl₂, Pt and Cr are paramagnetic; Fe, Co, Ni, Gd and alloys such as Alnico are ferromagnetic. The order of magnitude of χ for dia- and paramagnets in NCERT's table is 10⁻⁵; a question that gives χ = −2.6 × 10⁻⁵ wants the word diamagnetic and two properties, and a question that gives χ = 2.6 × 10⁻⁵ wants paramagnetic.

Diamagnetic substances are those which tend to move from a stronger to a weaker part of an external magnetic field. In other words, they are feebly repelled by a magnet, the opposite of what iron does. When a diamagnetic rod is placed in a field, the field lines are expelled from it and the field inside is slightly reduced; a freely suspended diamagnetic rod sets itself perpendicular to the field. Examples: bismuth, copper, lead, silicon, nitrogen at STP, water and sodium chloride.

Mechanism. Electrons in an atom orbit the nucleus and each orbit is a tiny current loop with a magnetic moment. In a diamagnetic atom the moments of the electrons cancel, so the atom has no net moment. When an external field is applied, the electrons whose orbital moment is along the field slow down and those whose moment is opposite to the field speed up (this is Lenz's law at the atomic level, the same physics as electromagnetic induction). The result is a small net induced moment opposite to the applied field, and the material is repelled. Because the effect is induced and does not depend on any pre-existing atomic moment, it is present in every material, but it is masked when the atoms have permanent moments, as in para- and ferromagnets.

Since thermal motion does not affect an induced moment, diamagnetism is nearly independent of temperature. That one line answers the frequent 1-mark question.

Superconductors. Some metals cooled to very low temperatures show perfect conductivity and perfect diamagnetism: the field lines are completely expelled, χ = −1 and μ_r = 0. This is the Meissner effect, and it is why a magnet can be levitated above a superconductor. NCERT mentions that superconducting magnets are used in the fastest trains. The value χ = −1 is the lower limit of the diamagnetic range, which is why the range is written −1 ≤ χ < 0.

Paramagnetic substances are those which get weakly magnetised when placed in an external magnetic field, in the same direction as the field. They tend to move from a weaker to a stronger part of the field, so they are feebly attracted by a magnet. The field lines are drawn slightly into a paramagnetic sample and the field inside is slightly larger than outside. Examples: aluminium, sodium, calcium, oxygen at STP and copper chloride.

Mechanism. Each atom (or ion or molecule) of a paramagnetic material has a permanent magnetic dipole moment of its own, because the electron moments do not cancel. Without a field, thermal agitation keeps these moments randomly oriented and the net magnetisation is zero. An applied field B₀ exerts a torque on each moment and tends to align it along the field; thermal motion fights this alignment. The balance gives a small net magnetisation along B₀. A stronger field or a lower temperature improves the alignment, so the magnetisation grows with B₀/T.

Curie's law. Experimentally, for not too strong fields and not too low temperatures, M = C B₀/T, equivalently χ = C μ₀/T, where C is the Curie constant of the material and T is the absolute temperature. The susceptibility of a paramagnet is therefore inversely proportional to temperature: cool the sample to half the absolute temperature and χ doubles. This is why a paramagnetic sample shows more magnetisation for the same field when cooled, a question asked directly in the exam.

Saturation. Curie's law cannot hold at very high fields or very low temperatures, because once every atomic dipole is aligned the magnetisation cannot grow any further. The magnetisation then saturates at M_s, the value when all moments point along the field. Note also the contrast with diamagnetism: diamagnetism comes from induced moments and ignores temperature; paramagnetism comes from permanent moments and is controlled by temperature.

Ferromagnetic substances get strongly magnetised when placed in an external field, in the direction of the field. They move from the weaker to the stronger region and are strongly attracted by a magnet; the field lines crowd inside the sample and the field inside can be hundreds or thousands of times the applied field. Examples: iron, cobalt, nickel, gadolinium and alloys like Alnico.

Mechanism: domains. As in a paramagnet, each atom has a permanent moment. In a ferromagnet, however, neighbouring atoms interact so strongly (the quantum-mechanical exchange interaction, which NCERT names but does not explain) that the moments of a large group of atoms align spontaneously. Such a region of uniform magnetisation is called a domain. A domain is about 1 mm across and contains about 10¹¹ atoms, so each domain has a large moment. In an unmagnetised sample the domains are oriented randomly and their moments cancel. When a field is applied, the domains aligned with the field grow at the expense of the others, and the domains also rotate towards the field. The whole sample then acts as a giant magnet, which is why χ is so large.

Hard and soft. In some ferromagnets the magnetisation persists after the field is removed; these are hard ferromagnets (Alnico, used for permanent magnets). In others the magnetisation disappears once the field is removed; these are soft ferromagnets (soft iron, used for electromagnets and transformer cores). The hysteresis (B-H) loop that quantifies retentivity and coercivity has been removed from the rationalised syllabus, but you should still know the two words hard and soft.

Curie temperature. Heating a ferromagnet increases thermal agitation and disrupts the domain ordering. Above a characteristic temperature T_c, the Curie temperature, the domain structure disintegrates and the material becomes paramagnetic. In the paramagnetic phase above T_c, the susceptibility follows χ = C/(T − T_c), which is Curie's law with the temperature measured from T_c. NCERT tabulates: cobalt 1394 K, iron 1043 K, Fe₂O₃ 893 K, nickel 631 K, gadolinium 317 K. An iron nail heated red hot is not attracted by a magnet, which is the everyday evidence.

NCERT's domain example (Example 5.10 in the older text) is worth reproducing: a cubic domain of iron of side 1 μm has volume 10⁻¹⁸ m³ = 10⁻¹² cm³, mass 7.9 × 10⁻¹² g (density 7.9 g cm⁻³), hence 7.9 × 10⁻¹² × 6.023 × 10²³/55 ≈ 8.65 × 10¹⁰ atoms. With 9.27 × 10⁻²⁴ A m² per atom, the maximum moment is 8.0 × 10⁻¹³ A m² and the maximum magnetisation is 8.0 × 10⁵ A m⁻¹, the same order as the M found in the solenoid example.