CBSE · Class 12 · Physics · Chapter 5

Magnetism and Matter — Formula Sheet

Board Formulas

12 formulas

1.Magnetic Field on the Axis of a Bar Magnet★
$B_{a x ia l} = \frac{μ _{0}}{4 π} \frac{2 m}{r ^{3}}$
$B_{a x ia l}$ : Magnetic field on the axis (T) · $m$ : Magnetic dipole moment (A m² or J T⁻¹) · $r$ : Distance from the centre of the magnet (m) · $μ_{0}$ : Permeability of free space = 4π×10⁻⁷ T m A⁻¹
Valid for a short magnet, i.e. r much larger than the magnet's length. B points along m (from the S-end to the N-end direction). Unit: tesla (T).
2.Magnetic Field on the Equatorial Line of a Bar Magnet
$B_{e q} = - \frac{μ _{0}}{4 π} \frac{m}{r ^{3}}$
$B_{e q}$ : Magnetic field on the equatorial line (T) · $m$ : Magnetic dipole moment (A m²) · $r$ : Distance from the centre of the magnet (m)
Magnitude is half the axial field at the same distance. The minus sign means B is antiparallel to m. Valid only for r much larger than the magnet's length.
3.Torque on a Magnetic Dipole in a Uniform Field★
$τ = m \times B, τ = m B sin θ$
$τ$ : Torque (N m) · $m$ : Magnetic dipole moment (J T⁻¹) · $B$ : Uniform magnetic field (T) · $θ$ : Angle between m and B
The torque tries to align m with B. In a uniform field the net force is zero, so the magnet only rotates. Unit: N m.
4.Potential Energy of a Magnetic Dipole★
$U_{m} = - m \cdot B = - m B cos θ$
$U_{m}$ : Magnetic potential energy (J) · $θ$ : Angle between m and B
Zero of energy is taken at θ = 90°. Minimum −mB at θ = 0° (stable), maximum +mB at θ = 180° (unstable). Unit: joule (J).
5.Work Done in Rotating a Magnetic Dipole
$W = m B (cos θ_{1} - cos θ_{2})$
$W$ : Work done (J) · $θ_{1}, θ_{2}$ : Initial and final angles between m and B
Work done by an external agent turning the dipole from θ₁ to θ₂; equals U(θ₂) − U(θ₁). From 0° to 180°, W = 2mB.
6.Gauss's Law for Magnetism
$\oint_{S} B \cdot d S = 0$
$B$ : Magnetic field (T) · $d S$ : Area element of the closed surface (m²)
Net magnetic flux through any closed surface is zero, because isolated magnetic poles (monopoles) do not exist. Magnetic field lines always form closed loops.
7.Magnetisation
$M = \frac{m _{n e t}}{V}$
$M$ : Magnetisation (A m⁻¹) · $m_{n e t}$ : Net magnetic moment of the sample (A m²) · $V$ : Volume of the sample (m³)
Net magnetic moment per unit volume of the material. Same unit as H: A m⁻¹.
8.Magnetic Intensity in a Solenoid
$H = n I$
$H$ : Magnetic intensity (A m⁻¹) · $n$ : Number of turns per unit length (m⁻¹) · $I$ : Current in the winding (A)
H inside a long solenoid depends only on turns per unit length and current, not on the core material. Unit: A m⁻¹.
9.Field Inside a Magnetised Material★
$B = μ_{0} (H + M), H = \frac{B}{μ _{0}} - M$
$B$ : Magnetic field (T) · $H$ : Magnetic intensity (A m⁻¹) · $M$ : Magnetisation (A m⁻¹)
H is set by external (free) currents; M is the material's response. B (in T) is the total field.
10.Magnetic Susceptibility
$M = χ H$
$χ$ : Magnetic susceptibility (no unit)
χ is dimensionless. Diamagnetic: χ small and negative (−1 ≤ χ < 0). Paramagnetic: χ small and positive. Ferromagnetic: χ very large and positive.
11.Permeability and Susceptibility★
$B = μ_{0} μ_{r} H = μ H, μ_{r} = 1 + χ$
$μ_{r}$ : Relative permeability (no unit) · $μ$ : Permeability of the material (T m A⁻¹)
Diamagnetic: μr slightly less than 1. Paramagnetic: μr slightly more than 1. Ferromagnetic: μr ≫ 1. μr and χ have no unit; μ has the unit of μ₀ (T m A⁻¹).
12.Curie's Law (Paramagnetism)
$χ = \frac{C μ _{0}}{T}$
$C$ : Curie constant of the material · $T$ : Absolute temperature (K)
Susceptibility of a paramagnet is inversely proportional to absolute temperature. Equivalent form: M = C B₀/T. Holds while the magnetisation is far from saturation. The rationalised NCERT text describes this temperature effect only in words, without the equation.