Formula sheet: Magnetism and Matter
Magnetism and Matter12 min readLesson 1 of 6
1.1 Dipole and field formulas
A bar magnet is treated as a magnetic dipole with magnetic moment m. Its field at a point on the axis at distance r is twice the field at the same distance on the equatorial line, and both fall as the inverse cube of the distance.
| Quantity | Expression |
|---|---|
| Axial field of a short dipole | B = (mu0 / 4 pi)(2m / r cubed) |
| Equatorial field of a short dipole | B = (mu0 / 4 pi)(m / r cubed) |
| Torque on a dipole in a field | tau = m x B, magnitude mB sin theta |
| Potential energy of a dipole | U = -m dot B = -mB cos theta |
| Work in turning from angle 1 to angle 2 | W = mB(cos theta1 - cos theta2) |
| Moment of a current loop | m = NIA |
1.2 Magnetic elements of the earth
The earth behaves like a bar magnet whose axis is tilted from the geographic axis. At a place, the earth's field B is resolved into a horizontal component B_H and a vertical component B_V, with declination D and angle of dip delta.
| Relation | Meaning |
|---|---|
| B_H = B cos delta | Horizontal component |
| B_V = B sin delta | Vertical component |
| tan delta = B_V / B_H | Angle of dip |
| B = square root of (B_H squared + B_V squared) | Total field |
1.3 Magnetisation relations
Magnetisation M is the magnetic moment per unit volume. The magnetic intensity H is defined through the field that would exist without the material, and the two combine to give the flux density inside the material.
| Relation | Meaning |
|---|---|
| M = m_net / V | Magnetisation of a sample |
| H = B_0 / mu0 | Magnetic intensity |
| B = mu0 (H + M) | Field inside the material |
| chi = M / H | Magnetic susceptibility |
| mur = 1 + chi | Relative permeability |
| mu = mu0 mur | Permeability |
| chi proportional to 1 / T | Curie law for a paramagnet |
1.4 Material data to remember
A diamagnetic material has small negative susceptibility, a relative permeability just below one, and is weakly repelled by a magnet. A paramagnetic material has small positive susceptibility, a relative permeability just above one, and is weakly attracted. A ferromagnetic material has a very large positive susceptibility, a relative permeability of the order of a thousand or more, and is strongly attracted. Iron, cobalt, nickel and gadolinium are ferromagnetic, and a ferromagnet becomes paramagnetic above its Curie temperature.