Illustrated Physics
physics, drawn
Concept

Field lines — where it appears

Curves drawn along a field's direction with their density standing for its strength, a convention that keeps three facts and discards the rest. They are not objects, do not move, and cannot be counted in three dimensions the way the drawing suggests — the density is a two-dimensional accident of the page.

Named by 12 essays across one field — each of them below, with the objects they name alongside it.

The field of a dipole. Field lines traced from a positive charge toward a negative one. Every line does eventually close on the negative charge, but the outer ones loop far outside any frame, so this picture is a crop rather than the whole field.

Field lines are a choice, not a discovery

Nothing in space is arranged in lines. The lines are a drawing convention — and an unusually good one, because three separate facts about the field survive the translation.

electromagnetism · The field concept The same field, sampled as arrows. The field at a grid of points, each arrow pointing the way a positive test charge would be pushed and scaled by the strength there.

The field before the lines were drawn on it

A field is a vector attached to every point of space. Drawing it as arrows on a grid is honest and ugly; drawing it as lines is beautiful and throws information away.

electromagnetism · The field concept A closed surface with the charge inside. Every field line from the enclosed charge crosses the surface exactly once on its way out, so the net flux counts the charge.

Counting what comes out, and never looking inside

Draw any closed surface. The field crossing it depends only on the charge enclosed — not on where that charge sits, not on its shape, not on anything outside.

electromagnetism · Gauss's law The field of a current loop, seen edge on. Magnetic field lines integrated from the Biot–Savart law for the current shown. Every line closes on itself: there is nowhere for one to start and nowhere for it to end, because no magnetic charge exists to end on.

The field with no ends, and the force that does no work

Magnetic field lines never start and never stop. That single absence is a law, it has survived every attempt to break it, and it makes the magnetic field a different kind of object from the electric one.

electromagnetism · Magnetism A loop leaving the field. A rectangular loop of wire 0.3 metres by 0.2 metres moving at 1.5 metres per second out of a region of magnetic field of 0.6 tesla directed into the page, marked with crosses. 0.08 metres of the loop's width is still inside the field. The induced current runs clockwise, and the force on the side that is in the field opposes the motion.

The field that makes the other, and only while it is changing

A magnet sitting next to a coil does nothing at all. Move it and a current flows. The law is not about the field but about its rate of change, and everything electrical since 1831 rests on that distinction.

electromagnetism · Induction The plane deleted, and one charge put in its place. A charge of 1 nC held 20 mm above an earthed conducting plane. The lines are traced through the field of the real charge plus an equal and opposite one at the mirror position, and then cut at the plane, because below it there is metal and no field whatever. Nothing in the tracing knows about the surface: each line follows the local field direction and stops where it arrives. That every one of them arrives perpendicular — the worst departure among the 9 drawn is 2.2° away from square — is the boundary condition showing itself rather than a rule imposed on the drawing. The image charge is drawn faint because it is not there: it is a way of writing a function that happens to satisfy the equation and the boundary values, which by the uniqueness theorem makes it the field and not a model of the field.

The charge that has to be somewhere else

Hold a charge above an earthed metal sheet and the field above it is exactly the field of two charges — the real one and an imaginary partner buried at the mirror position. The partner is not an analogy or an approximation. It is a legal guess, and a legal guess is a proof.

electromagnetism · Conductors A disc that generates 0.785 V with no changing flux anywhere. The force per unit charge on a carrier in a conducting disc of radius 100 mm spinning at 3000 revolutions a minute in an axial field of 0.5 tesla, against how far out the carrier is. Each carrier is moving through the field at ωr, so v × B pushes it radially with a force per unit charge of ωrB — zero at the axle and largest at the rim. The area under the line is the potential difference between the axle and the rim, which is ½BωR² = 0.7854 volts. Nothing in this calculation mentions a loop, a flux or a rate of change, and the flux through the circuit formed by the disc, the contacts and the external wire does not change at all while the machine runs.

The rule that is two laws wearing one coat

The flux rule folds two entirely different pieces of physics into one number, and they agree exactly, which is why nobody notices there are two. A disc spinning in a steady field generates a voltage with no changing flux anywhere, and the folding comes apart.

electromagnetism · Induction The shell of news, and the kink inside it. A charge that was moving at 0.6c to the right, stopped over a short interval, and has been at rest ever since. Outside the sphere of radius ct nothing has heard: the field there still points at the position the charge would have reached had it carried on, marked ahead of it. Inside, the field is that of a charge at rest. In between is a shell one deceleration-time thick, and across it the field line has to bend, because a field line cannot simply stop in empty space — the two ends are joined here to 8.5e-14 pixels. That bend is transverse to the radius, and it is the radiation. It is not an extra thing the charge emitted; it is the join between two static fields that do not line up, and it travels outward at c because that is where the news front is.

The field that points where the charge is now

The field here was set by what the charge was doing a distance over c ago, so it ought to point at where the charge used to be. For a charge moving steadily it points at where the charge is — not approximately, exactly — and nothing has outrun light. What breaks the arrangement is a change of motion, and the break is the whole of radiation.

electromagnetism · Retardation Every line that goes in has to come out. Field lines of a 5:1 solenoid in the plane through its axis, each traced by stepping along the local direction of a field summed turn by turn from Biot–Savart. Inside the winding they are parallel and evenly spaced, which is the picture the textbook argument is about. Outside they are not absent: they are spread over the whole of the rest of space, which is why the field there is small — 1.60e-2 of the centre value at 2 radii off the axis — and why it cannot be zero. A line has no end, so every one of the lines through the bore returns outside, and a field with no outside would be a field whose lines stop.

The field outside the solenoid, which is not zero

Ampère's law says the field outside a solenoid vanishes, and every step of that argument is exact — for a winding of infinite length. A real one is a bar magnet seen from outside, its external field falls as the inverse square of its length rather than to nothing, and the "exactly zero" that makes the derivation so satisfying is the one part of it a laboratory cannot have.

electromagnetism · Ampere law A refraction with no wave in it. Field lines crossing the boundary between two dielectrics whose permittivities differ by a factor of 4, at incidences of 12°, 26°, 40°, 54°, 68°. Nothing is oscillating and nothing is travelling: these are static fields, and the only inputs are that the component of E along the surface is the same on both sides and that the component of D across it is. Dividing one condition by the other gives the ratio of the tangents of the two angles, and it comes out equal to the ratio of the permittivities — measured off the constructed directions as 4.000, 4.000, 4.000, 4.000, 4.000, against 4. The lines bend towards the surface on the side with the larger permittivity, which is the opposite sense from a light ray entering glass, and the chart on the right shows the whole relation: it is a version of Snell's law with the sines replaced by tangents, and there is no speed anywhere in it. Two things follow that are worth carrying. A field line meeting a surface at grazing incidence stays nearly parallel to it whatever the materials, so the ratio matters least where the field is largest along the surface. And in the limit of a very large ratio every line comes out very nearly perpendicular on the low side, which is the electrostatic ancestor of a conductor's boundary condition — a conductor is the ratio taken to infinity.

A refraction with no wave in it

A field line crossing from one dielectric into another bends, by a law that looks exactly like Snell's with the sines replaced by tangents. Nothing is oscillating, nothing is travelling, and no speed appears anywhere in the derivation — only the two conditions that say what a boundary may and may not do to a field.

electromagnetism · Dielectrics Four places where three charges' fields cancel. Field lines in the plane of three equal positive point charges at the corners of an equilateral triangle, with every point where the field is zero marked. There are four: one at the centre, and three more on the lines from the centre to the middle of each side, 0.164 of a side from the centre. Four is (n − 1)² for three charges, the largest number Maxwell conjectured any three charges could make. Each zero is a saddle: along some directions the field pushes a test charge back and along others away, so none of them can hold anything still.

The zeros a handful of charges can make

Between two equal charges there is a point where their fields cancel. With three there can be two such points, or one, or four. Every such point is a saddle, never a place where a charge could rest. In two dimensions the count is fixed by algebra: n equal line charges make exactly n − 1 zeros, the roots of a polynomial's derivative. In three dimensions nobody knows the answer. In 1873 Maxwell conjectured that n point charges can make at most (n − 1)² zeros. Three charges on a triangle make exactly four, and the conjecture has not been proved even for three charges.

electromagnetism · The field concept Charges released on a field line do not stay on it. A positive and a negative charge (left and right) with their field lines (thin), and the paths of a light positive test charge released at rest from four points 0.32 charge-separations from the positive charge, at 20°, 40°, 55°, 70° from the axis (thick), each beside the field line through its own starting point (dashed). Each path sets off exactly along its line — at rest, the acceleration is the only thing with a direction — and then swings wide of it, because the charge carries its velocity round the bend while the line turns. All four miss the negative charge and leave the frame, never to return, although the field line from every starting point ends on it.

The path a charge takes is not its field line

A field line is drawn by stepping along the direction of the force, and it is natural to read it as the route a charge would take. It is not. A field line is the curve whose direction is the force; a charge's path is the curve whose bending is the force, and the two agree only where the charge starts from rest. A charge released beside a dipole sets off along its line and then swings wide of it, carried by the speed it has gained; a charge launched round a point charge circles it on an ellipse that crosses every line. Only two things make a charge keep to its line: enough friction that it never has a speed of its own, or a magnetic field.

electromagnetism · The field concept

Named alongside it

The objects these essays reach for when they reach for this one.

Electric fieldFluxSuperpositionThe inverse-square lawMagnetic fieldSymmetryBoundary conditionsConductorElectromagnetic inductionEmfGauss's lawLenz's law

All concepts