Magnetism and Matter - Quick Revision
The bar magnet as a dipole
- A bar magnet behaves like a magnetic dipole; its field lines form continuous closed loops (no isolated poles / monopoles).
- Equivalent to a solenoid: magnetic moment m = N I A.
- Cutting a magnet gives two smaller magnets, never an isolated pole.
Dipole in a uniform field
- Net force is zero; torque tau = m x B, magnitude tau = m B sin(theta).
- Potential energy U = -m.B = -m B cos(theta) (zero taken at theta = 90).
- Stable at theta = 0 (U = -mB, minimum); unstable at theta = 180 (U = +mB).
- Work to rotate: W = m B (cos theta_1 - cos theta_2).
Field of a short bar magnet (r >> l)
- Axial: B_A = (mu0/4 pi)(2m/r^3), along m.
- Equatorial: B_E = (mu0/4 pi)(m/r^3), opposite to m - half the axial value.
- mu0/4 pi = 10^-7 T m/A.
Gauss's law for magnetism
- Net magnetic flux through any closed surface is zero: integral of B.dS = 0.
- Reflects the absence of magnetic monopoles.
Magnetisation and intensity
- Magnetisation M = m_net/V (A/m).
- B = mu0 (H + M); in a solenoid H = n I (independent of the core).
- M = chi H, mu_r = 1 + chi, mu = mu0 mu_r.
Classifying materials
- Diamagnetic: chi small and negative, mu_r < 1, repelled, move to weak field; superconductor is perfect (chi = -1, Meissner effect).
- Paramagnetic: chi small and positive, mu_r slightly > 1, weakly attracted; obeys Curie law M proportional to B0/T.
- Ferromagnetic: chi large positive, mu_r >> 1, strongly attracted, domains; becomes paramagnetic above the Curie temperature.
Common traps
- Equatorial field is HALF the axial field, not equal.
- H = nI does NOT depend on the core; B does (B = mu_r mu0 H).
- Stable equilibrium is theta = 0 (U most negative), not theta = 90.
- Net magnetic flux through any closed surface is always zero (monopoles do not exist).