Magnet field lines - 2D dipole map

The field of a magnet

Far from a bar magnet its field is that of a magnetic dipole with moment m: B(r) = (μ₀ / 4π) [3 (m·&rhat;) &rhat; - m] / r³. On the axis this is B = μ₀ m / (2π r³), and on the equator it is half as strong and points the other way. So the field falls as 1/r³, much faster than the 1/r² of an electric charge. Field lines are curves tangent to B; they leave the north pole and re-enter the south pole, and they never cross because B has one direction at each point. Outside the sources, curl B = 0, so B = -∇Φ_M, which is why the line integral of B around any loop that encloses nothing is zero.

Forces between magnets

A second dipole in the field of the first has energy U = -m₂·B₁ and feels a torque τ = m₂ × B₁ that turns it along the field, plus a force F = ∇(m₂·B₁). Two coaxial magnets aligned head to tail attract with F = 3 μ₀ m₁ m₂ / (2π d&sup4;), falling as the fourth power of the distance. The map colours the field strength on a logarithmic scale and the needles are compass probes.

Honest simplifications: a bar magnet is treated as an ideal point dipole, so the field is exact only beyond about two magnet lengths and the lines near each magnet are indicative; the picture is a cut through a 3D field in the plane containing both moments; no magnetic materials, saturation or demagnetisation, and the force on magnet 2 is computed only from the field of magnet 1 (as the force on a rigid dipole).

• Choose Repel (N-N facing) and Attract (N-S facing), and compare the field line pattern between the magnets.
• Drag magnet 2 away and watch the force fall as 1/d to the fourth.
• Rotate magnet 2 and read the torque graph: it is zero where the magnet lies along the field, a stable point.
• Turn on the field strength map to see the saddle point of zero field between two repelling magnets.