Magnetic Fields & Field Representation

Electrostatics defines E by the force on a charge at rest, E = F/q. That test fails outright for magnetism, because a stationary charge in a magnetic field feels nothing. So B is defined by motion. Fire a test charge q through a point with speed v, swinging the direction of v through every orientation, and two things emerge. There is exactly one axis along which the force vanishes: that axis is the line of B, and a compass needle's north end picks which of the two senses along it to call the direction. Across that axis the force is largest, and the ratio Fₘₐₓ/(|q| v) comes out the same whatever test charge you use — that ratio is the magnitude B, in tesla. An electron at 2.0 × 10⁶ m s⁻¹ crossing a 0.050 T field feels 1.6 × 10⁻¹⁴ N, tiny in absolute terms and still about 1.8 × 10¹⁵ times its own weight. The force law itself is the next topic; what matters here is that this definition makes B a property of the point, not of the charge that probed it.

A field line is drawn tangent to B at every point along it, so it gives the field's direction there and only there; the line's shape a metre away tells you nothing about the field where you are standing. Two lines may never cross, because a crossing would hand one point two field directions at once. Spacing carries magnitude: lines are drawn so that the number crossing a fixed area is proportional to B, which is why they crowd near a pole and thin out far from it. Flux makes that exact, Φ_B = ∫ B⋅dA. A uniform 0.40 T field through a flat 25 cm² loop held face-on gives Φ_B = 0.40 × 25 × 10⁻⁴ = 1.0 × 10⁻³ Wb, and tilting the loop's normal by 60° to the field halves it. On paper a field lying in the page is drawn as arrows, and one perpendicular to the page as ⊙ for out of the page, the tip of an arrow, and ⊗ for into it, the tail feathers. Nothing in the picture is countable: there is no true number of lines, only flux.

Electric field lines begin on positive charge and end on negative charge, which is what ∮E⋅dA = q_enc/ε₀ says in pictures: enclose charge and the surface leaks flux. Magnetic lines do neither. Follow one out of a bar magnet's north end, round through the air and in at the south end — and it keeps going, up through the iron itself from south back to north, closing the loop. A line that closes on itself has no end at which flux could be created, so every line leaving a closed surface must come back in through it somewhere else. That is a law of its own: ∮B⋅dA = 0 through every closed surface, whatever sits inside it. Wrap a surface around the north end alone and the flux out through the air is cancelled exactly by flux entering where the magnet's body crosses the surface. Two consequences are worth carrying forward. The law is exact rather than a weak-field or high-symmetry approximation, and any two surfaces sharing one rim pass the same flux — which is why the loop in the third example can be capped with a flat disc or with a dome and give the same answer.

Magnetic fields superpose: the field at a point is the vector sum of what every source contributes, each computed as though the others were absent. Work in components. A compass in an 18 μT northward horizontal Earth field, with a magnet adding 12 μT due east, sits in a resultant of √(18² + 12²) = 21.6 μT pointing arctan(12/18) = 33.7° east of north. Swing the magnet round so its 12 μT points due south and the components subtract along one line: 18 − 12 = 6 μT north, and the needle still points north but is held only feebly. Strengthen it to 24 μT south and the resultant is 6 μT south — the needle turns right round. Between those two settings lies a null point where the two cancel exactly, the resultant is zero, and the needle has no preferred direction at all. That superposition is what the figure below draws, and it is why a compass indoors is worth so little.

A compass needle is a small dipole free to turn about a vertical pivot, so it settles along the horizontal component of the field and ignores the vertical component entirely. Three consequences are worth stating. It reports a direction, never a magnitude: it cannot tell 20 μT from 200 μT. It does not point at the geographic pole — the angle between the field's horizontal part and true north is the declination, a few degrees across most of Europe. And most of the field is invisible to it: where the total is 49 μT at a dip of 66° below horizontal, only 20 μT lies in the horizontal plane. The instrument becomes quantitative only as a comparator. Hold a known horizontal field Bₕ, apply an unknown transverse field Bₜ at right angles, and the needle settles where tan θ = Bₜ/Bₕ; a deflection of 34.0° in a 19.0 μT Earth field gives Bₜ = 19.0 × tan 34.0° = 12.8 μT. The dip angle needs a second instrument, a dip circle.

Set the two fields side by side. Sources: E comes from charge, B comes from charge in motion — currents, and the intrinsic spin of electrons, which is what a permanent magnet is built from. What they act on: E pushes any charge, at rest or moving, while B acts only on a moving charge, and only on the part of that motion across the field. Direction: the electric force lies along E, or against it for a negative charge, whereas the magnetic force is perpendicular to both v and B, so it never lies along a field line. Work: an electric force can speed a charge up, a magnetic force never can, because a force perpendicular to v does no work. Topology: E lines start and stop on charge, B lines close. Even the units record the difference — E is quoted in N C⁻¹, while 1 T = 1 N A⁻¹ m⁻¹, a force per unit current per unit length, because it takes a moving charge to feel B at all.