Magnetism and Matter notes: ISC Class 12 Physics

1. The bar magnet, its equivalent solenoid and its field lines

Field lines of a bar magnet run from N to S outside and from S to N inside: every line is a closed loop; lines never cross. The magnetic moment $\vec m$ points from S to N (A m²).
A current-carrying solenoid has the same field pattern: it is the equivalent solenoid of the magnet, with $m = NIA$ . In a magnet, aligned atomic current loops act like the turns of the solenoid.
Electric lines start on +q and end on −q, and between the charges $\vec E$ is opposite to $\vec p$ ; magnetic lines have no ends and inside the magnet $\vec B$ is along $\vec m$ . No magnetic monopole has been found, so the net magnetic flux through any closed surface is zero.

Formulas

Moment of a solenoid
$m = NIA$

Key terms

Magnetic monopole: A hypothetical isolated north or south pole; none has been observed, so magnetic field lines have no ends.
Equivalent solenoid: A current-carrying solenoid whose field pattern and moment NIA match those of a given bar magnet.

2. Field of a magnetic dipole; torque on a magnet

Magnet of magnetic length 2l, moment m, point at distance d from its centre (formulae only; derivations not required):
- end-on (on the axis): $B = \dfrac{\mu_0}{4\pi}\dfrac{2md}{(d^2 - l^2)^2}$ , along $\vec m$ ;
- broadside-on (on the perpendicular bisector): $B = \dfrac{\mu_0}{4\pi}\dfrac{m}{(d^2 + l^2)^{3/2}}$ , opposite to $\vec m$ .
Short magnet ( $d \gg l$ ): $\dfrac{\mu_0}{4\pi}\dfrac{2m}{d^3}$ and $\dfrac{\mu_0}{4\pi}\dfrac{m}{d^3}$ ; both $\propto 1/d^3$ ; end-on is twice broadside-on at the same d. $\mu_0/4\pi = 10^{-7}$ T m A⁻¹.
In a uniform field: no net force, torque $\vec\tau = \vec m \times \vec B$ , $\tau = mB\sin\theta$ ; stable at θ = 0°, unstable at θ = 180°, largest (mB) at 90°.

end-on: 2m/d³
broadside-on: m/d³

Doubling d divides either field by 8.

Formulas

Field of a bar magnet
$B_\text{end-on} = \dfrac{\mu_0}{4\pi}\dfrac{2md}{(d^2 - l^2)^2},\qquad B_\text{broadside-on} = \dfrac{\mu_0}{4\pi}\dfrac{m}{(d^2 + l^2)^{3/2}}$
Torque on a magnet
$\vec\tau = \vec m \times \vec B$

Key terms

End-on position: A point on the axis of the magnet, produced; the field there points along m.
Broadside-on position: A point on the perpendicular bisector of the magnet; the field there points opposite to m.

3. Magnetic flux and flux density

Magnetic flux through a flat surface in a uniform field: $\Phi = \vec B \cdot \vec A = BA\cos\theta$ , θ between $\vec B$ and the normal. A scalar; unit weber (1 Wb = 1 T m²).
Largest ( $BA$ ) when the surface faces the field; zero when the field lies along the surface; negative when B passes through the other face.
$B = \Phi/A$ is the flux density: unit tesla, 1 T = 1 Wb m⁻². Gauss: 1 G = 10⁻⁴ T. Earth’s field is about 0.5 G; a strong laboratory magnet about 1 T.

Φ = BA cos θ

Beyond 90° the field passes through the other face: the flux is negative.

Formulas

Magnetic flux
$\Phi = BA\cos\theta$
Units
$1\ \text{T} = 1\ \text{Wb m}^{-2} = 10^{4}\ \text{G}$

Key terms

Magnetic flux: The scalar product of B with the area vector of a surface, BA cos θ; unit weber.
Gauss: A non-SI unit of magnetic field: 1 G = 10⁻⁴ T.

4. Magnetic properties of materials

Magnetisation $M = m/V$ (A m⁻¹); magnetic intensity $H$ (A m⁻¹), $H = nI$ in a long solenoid; $B = \mu_0(H + M)$ .
Susceptibility $\chi = M/H$ (no unit); relative permeability $\mu_r = 1 + \chi$ ; $B = \mu_0\mu_r H$ ; permeability $\mu = \mu_0\mu_r$ .
Diamagnetic: χ small, negative, μr slightly < 1; induced moments oppose the field; pushed towards weaker field; almost independent of temperature (bismuth, copper, water).
Paramagnetic: χ small, positive, μr slightly > 1; permanent moments partly aligned; pulled towards stronger field; Curie’s law $\chi = C/T$ (aluminium, sodium, oxygen).
Ferromagnetic: χ and μr very large; moments aligned in domains; above the Curie temperature $T_C$ it becomes paramagnetic (iron about 1043 K, nickel about 631 K, gadolinium about 293 K).

Table comparing diamagnetic, paramagnetic and ferromagnetic substances by atomic moments, susceptibility, relative permeability, behaviour in a non-uniform field, effect of temperature and examples
	Diamagnetic	Paramagnetic	Ferromagnetic
Atomic moments	none permanent; induced, opposing B	permanent, randomly oriented	permanent, aligned in domains
χ	small, negative ( $\sim -10^{-5}$ )	small, positive ( $10^{-5}$ to $10^{-3}$ )	large, positive (hundreds to thousands)
μr	slightly below 1	slightly above 1	much greater than 1
In a non-uniform field	pushed to weaker field	pulled to stronger field	strongly pulled in
Effect of temperature	almost none	χ = C/T (Curie’s law)	paramagnetic above $T_C$
Examples	bismuth, copper, water	aluminium, sodium, oxygen	iron, nickel, cobalt

χ = C/T, C = 0.112 K

Curie’s law: χT stays constant.

Formulas

B, H and M
$B = \mu_0(H + M),\qquad M = \chi H,\qquad \mu_r = 1 + \chi$
Curie’s law
$\chi = \dfrac{C}{T}$

Key terms

Susceptibility: Ratio M/H of the magnetisation of a material to the magnetic intensity; negative for diamagnets.
Curie temperature: The temperature above which a ferromagnet becomes paramagnetic, because thermal agitation destroys the alignment within its domains.
Domain: A small region of a ferromagnet in which the atomic moments are spontaneously aligned.

5. Electromagnets, the B–H loop and the choice of materials

Electromagnet: a coil on a soft-iron core, $B \approx \mu_0\mu_r nI$ ; stronger with more turns per unit length, more current and a core of higher permeability (and a small air gap), until the core saturates.
B–H loop (qualitative): B rises to saturation; at H = 0 a field B, the retentivity, remains; a reverse H, the coercivity, is needed to make B zero. B lags behind H (hysteresis); the loop area measures the energy lost as heat per cycle.
Soft iron: narrow loop, low coercivity, high permeability, slightly higher retentivity than steel. Steel: wide loop, high coercivity.
Choice: permanent magnets — high retentivity and high coercivity (steel, alnico); electromagnet cores — high permeability, low retentivity and coercivity (soft iron); transformer cores — high permeability, narrow loop (laminated soft iron, silicon steel).

Qualitative: real coercivities differ by a factor of a thousand or more.

Table of the choice of magnetic materials: permanent magnets need high retentivity and coercivity, as in steel or alnico; electromagnet cores need high permeability and low retentivity and coercivity, as in soft iron; transformer cores need high permeability and a narrow loop, as in laminated soft iron or silicon steel
Use	Needs	Typical material
Permanent magnet	high retentivity, high coercivity	steel, alnico, ferrites
Electromagnet core	high permeability, low retentivity, low coercivity	soft iron
Transformer core	high permeability, low coercivity, narrow loop (small loss)	soft iron or silicon steel, laminated

Formulas

Electromagnet
$B \approx \mu_0\mu_r nI$

Key terms

Retentivity: The flux density remaining in a ferromagnet when H is reduced to zero after saturation.
Coercivity: The reverse magnetic intensity needed to reduce B in a magnetised ferromagnet to zero.
Hysteresis: The lagging of B behind H when a ferromagnet is taken round a cycle of magnetisation.

Exam tips

Field of a magnet: name the position and give the direction (end-on along m, broadside-on opposite to m).
Flux: θ is measured from the normal; 1 T = 1 Wb m⁻² = 10⁴ G.
Materials: classify by the sign and size of χ; Curie’s law needs kelvin.
B–H loop: permanent magnets need high coercivity; cores need low coercivity and a narrow loop.

1. The bar magnet, its equivalent solenoid and its field lines

Formulas

Key terms

2. Field of a magnetic dipole; torque on a magnet

Formulas

Key terms

3. Magnetic flux and flux density

Formulas

Key terms

4. Magnetic properties of materials

Formulas

Key terms

5. Electromagnets, the B–H loop and the choice of materials

Formulas

Key terms

Exam tips

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