CBSE Class 12 Physics Magnetism and Matter Notes Set 03

1. The Bar Magnet

A bar magnet is a permanent magnet in the shape of a rectangular bar. Its magnetism appears strongest near its two ends, called the north (N) and south (S) poles.
1. Like poles repel and unlike poles attract each other.
2. The two poles always occur together. If a bar magnet is cut into pieces, every piece is a complete magnet with its own N and S poles. An isolated magnetic pole (a magnetic monopole) has never been found.
3. Iron filings sprinkled on a sheet placed over a bar magnet arrange themselves along curves that show the pattern of the magnetic field. A small compass needle placed at any point sets itself along the field at that point.

2. Magnetic Field Lines

A magnetic field line is an imaginary curve drawn so that the tangent at any point gives the direction of the magnetic field B at that point.

Properties of magnetic field lines:
1. They form continuous closed loops. Outside a magnet they run from the N pole to the S pole; inside the magnet they run from the S pole to the N pole.
2. Two field lines never cross, because the field has only one direction at any point.
3. The number of lines crossing a unit area held normal to the lines indicates the strength of the field: where the lines are crowded (near the poles) the field is strong.
4. Field lines do not represent lines of force on a moving charge; the force on a moving charge is perpendicular to B.

Comparison with electric field lines: Electrostatic field lines start on positive charges and end on negative charges, so they never form closed loops. Magnetic field lines have no starting or ending points, because there are no magnetic monopoles.

3. Bar Magnet as an Equivalent Solenoid (Qualitative)

The field lines of a bar magnet and of a current-carrying solenoid are almost identical outside them (see the figure above). A solenoid behaves like a bar magnet:
1. The end where the current appears anticlockwise (seen from outside) acts as a N pole; the end where it appears clockwise acts as a S pole.
2. A freely suspended solenoid aligns itself like a magnetic needle, and two solenoids attract or repel each other like two bar magnets.
3. Each turn of the solenoid is a small current loop, i.e. a magnetic dipole. The magnetic moments of all the turns add up along the axis, so the whole solenoid has a net magnetic moment like that of a bar magnet.
This suggests that the magnetism of a bar magnet also comes from tiny circulating currents (of electrons) in its atoms.

4. Magnetic Dipole Moment of a Bar Magnet

A bar magnet behaves as a magnetic dipole. Its magnetic dipole moment m is a vector directed from its S pole to its N pole (inside the magnet). SI unit of m: A m² (also J T⁻¹).
For a solenoid of N turns, each of area A, carrying current I, m = NIA, directed along the axis.

Field of a bar magnet at large distances (results, qualitative):
1. At large distances, the field of a short bar magnet has the same form as the electric field of an electric dipole, with m in place of p and μ₀ in place of \( \frac{1}{\varepsilon_0} \).
2. On both the axial line and the equatorial line, B is proportional to \( \frac{m}{r^3} \).
3. At the same distance r, the field on the axial line is twice the field on the equatorial line. On the axial line B is along m; on the equatorial line it is opposite to m.

5. Magnetic Dipole in a Uniform Magnetic Field (Qualitative)

1. In a uniform field the forces on the two ends of a bar magnet are equal and opposite, so the net force is zero.
2. These forces form a couple, so the magnet experiences a torque that tends to turn m into the direction of B. The torque has the same form as for a current loop in Chapter 4, \( \vec{\tau} = \vec{m} \times \vec{B} \).
3. The torque is zero when m is parallel to B (stable equilibrium) or antiparallel to B (unstable equilibrium), and greatest when m is perpendicular to B.
4. This is why a compass needle, slightly disturbed from the field direction, swings back and oscillates about it before coming to rest along B.

6. Gauss's Law for Magnetism

The net magnetic flux through any closed surface is always zero.
\( \oint \vec{B} \cdot d\vec{S} = 0 \)

1. Every field line that enters a closed surface also leaves it, because magnetic field lines are closed loops (in the figure, the lines entering the dashed surface inside the magnet equal the lines leaving it from the N pole).
2. Gauss's law for magnetism means that isolated magnetic poles (monopoles) do not exist. The simplest magnetic element is a dipole or a current loop.
3. Compare with electrostatics: \( \oint \vec{E} \cdot d\vec{S} = \frac{q_{\text{enclosed}}}{\varepsilon_0} \), which can be non-zero because isolated charges exist.
4. Magnetic flux φ_B = BA cos θ through an open surface can be non-zero. SI unit of magnetic flux: weber (Wb) = T m².

7. Magnetisation and Magnetic Intensity

Every atom behaves as a tiny magnetic dipole because of the orbital and spin motion of its electrons. When a material is placed in a magnetic field, these atomic moments respond and the material itself becomes magnetised.

(i) Magnetisation (M): the net magnetic moment per unit volume of the material.
M = \( \frac{m_{\text{net}}}{V} \). It is a vector. SI unit: A m⁻¹.

(ii) Magnetic intensity (H): For a long solenoid with n turns per unit length carrying current I, the field in vacuum (air) inside is B₀ = μ₀nI. The magnetic intensity is defined by
H = \( \frac{B_0}{\mu_0} \) = nI. SI unit: A m⁻¹ (the same as M).
H depends only on the external source (the current), not on the material inside.

(iii) Total field inside a material: If the solenoid is filled with a material of magnetisation M,
B = μ₀(H + M)
The first part comes from the current in the windings and the second from the magnetised material. μ₀ = 4π × 10⁻⁷ T m A⁻¹ is the permeability of free space.

8. Magnetic Susceptibility and Permeability

For most materials M is proportional to H:
M = χH, where χ is the magnetic susceptibility, a dimensionless number that tells how strongly the material responds to a field.
Then B = μ₀(H + χH) = μ₀(1 + χ)H = μ₀μ_rH = μH
where
1. Relative permeability: μ_r = 1 + χ (dimensionless)
2. Permeability of the material: μ = μ₀μ_r = μ₀(1 + χ), unit T m A⁻¹ (or H m⁻¹)
3. Inside a long solenoid filled with the material, B = μ_rB₀ = μ₀μ_rnI.
4. For vacuum, χ = 0 and μ_r = 1.

9. Classification of Magnetic Materials

Materials are classified by the sign and size of χ: diamagnetic (χ small and negative), paramagnetic (χ small and positive) and ferromagnetic (χ large and positive).

The figure shows (with exaggerated effects) how a sample placed in a uniform field changes the field lines: a diamagnetic sample pushes the lines out, a paramagnetic sample draws a few more lines in, and a ferromagnetic sample crowds them strongly through itself.

10. Diamagnetism

1. Diamagnetic substances are weakly repelled by a magnet. In a non-uniform field they tend to move from the stronger to the weaker part of the field.
2. Their atoms have no net permanent magnetic moment. An external field changes the orbital motion of the electrons and induces a small magnetic moment opposite to the field (an effect like Lenz's law), so M is opposite to H.
3. χ is small and negative (about −10⁻⁵), so μ_r is slightly less than 1. Field lines are slightly expelled from the material.
4. A rod of diamagnetic material suspended in a uniform field turns to lie perpendicular to the field. A diamagnetic liquid in one arm of a U-tube placed between the poles of a magnet is pushed down (its level falls).
5. Diamagnetism is present in all substances, but it is noticed only where other effects are absent.
Examples: bismuth, copper, lead, silicon, nitrogen (at STP), water and sodium chloride.
6. Superconductors (some metals cooled to very low temperatures) are perfect diamagnets: χ = −1, μ_r = 0, and the field lines are completely expelled. This is the Meissner effect; it is used, for example, in magnetically levitated trains.

11. Paramagnetism

1. Paramagnetic substances are weakly attracted by a magnet. In a non-uniform field they tend to move from the weaker to the stronger part of the field.
2. Their atoms have a small permanent magnetic moment. In the absence of a field these moments point in random directions because of thermal agitation, so the net moment is zero. An external field partly lines them up along the field.
3. χ is small and positive (about 10⁻⁵ to 10⁻³), so μ_r is slightly greater than 1. Field lines are drawn slightly into the material.
4. A rod of paramagnetic material suspended in a uniform field turns to lie parallel to the field; a paramagnetic liquid in a U-tube rises in the arm placed between the poles.
5. Higher temperature increases the random thermal motion, so the alignment, and hence the magnetisation produced by a given field, becomes weaker.
Examples: aluminium, sodium, calcium, oxygen (at STP) and copper chloride.

12. Ferromagnetism

1. Ferromagnetic substances are strongly attracted by a magnet and move from weaker to stronger regions of a non-uniform field.
2. Their atoms have permanent magnetic moments that interact with one another and line up in a common direction over small regions called domains. A domain is about 1 mm across and contains about 10¹¹ atoms.
3. In an unmagnetised sample the domains point in random directions, so the net magnetisation is zero. In an external field, domains aligned with the field grow and the others turn towards the field, giving a very large magnetisation along the field.
4. χ is large and positive (of the order of 10³ or more), so μ_r ≫ 1 and the field lines crowd strongly into the material.
Examples: iron, cobalt, nickel, gadolinium and many of their alloys.

13. Comparison of Dia-, Para- and Ferromagnetic Materials

14. Quick Formula Recap