Illustrated Physics
physics, drawn
Series

The field concept — the series

7 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 1 · electromagnetism
  2. 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.

    part 2 · electromagnetism
  3. Equipotentials, with the field lines that cross them. Contours of constant potential, traced by marching squares, with field lines traced along the gradient of the same potential. The two families meet at right angles everywhere, which is a consequence of the field being the gradient rather than a property of the drawing.

    The attraction that needs no charge

    Gauss's law says nothing comes out of a neutral molecule, and yet water's field one nanometre away reaches 1.1 × 10⁸ V/m. What survives when the monopole vanishes is a separation, and every step down the tower of falloffs below it is paid for with one more order of cancellation.

    part 3 · electromagnetism
  4. A linear quadrupole, and the cone on which it vanishes. Equipotentials of a linear quadrupole — charges +q, −2q, +q in a line, a distance d apart — in a plane containing the line, positive in one colour and negative in the other. The total charge is zero and so is the dipole moment, so far away the potential falls as 1/r³ with the angular pattern 3cos²θ − 1: positive along the axis, negative around the waist, and zero on a cone. Found on circles of growing radius, the zero of the exact potential lies at 50.89° at 2d, 53.76° at 4d, 54.49° at 8d, 54.67° at 16d, closing on arccos(1/√3) = 54.74°, the dashed lines.

    The shape that takes five numbers

    A charge distribution's net charge is one number and its dipole three. The next term, the quadrupole, is where the distribution's shape first reaches the far field, and it is a table of nine numbers of which only five survive: two for the shape and three for which way it points. Its field vanishes on a cone at 54.74° — the same angle at which a spinning sample makes a nuclear magnetic resonance spectrum sharp.

    part 4 · electromagnetism
  5. 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.

    part 5 · electromagnetism
  6. 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.

    part 6 · electromagnetism
  7. Two fields that part company in the middle. The direction of the electric field of two opposite charges, on the left, and of the magnetic field of a loop of current cut across, on the right, each with a dipole moment pointing up the page. Outside, the arrows of the two pictures are nearly the same and become identical with distance. In the middle, drawn darker, they point opposite ways: the field between the two charges runs from the positive to the negative, downward, against the moment, and the field through the loop runs upward, with it. Field lines of the magnetic dipole close through the middle; those of the electric dipole begin and end there. No measurement made from a long way off can tell them apart.

    Two dipoles that agree everywhere but inside

    A pair of opposite charges and a small loop of current make the same field at any distance worth measuring. Between the charges that field points one way, and through the loop it points the other. Shrink both to a point and the disagreement does not vanish: it becomes a term concentrated at the origin, and hydrogen's 21-centimetre line depends on it. The level order of the hydrogen atom says which kind of dipole a proton is.

    part 7 · electromagnetism

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