Illustrated Physics
physics, drawn
Concept

Field momentum — where it appears

The momentum carried by an electromagnetic field itself, equal to the Poynting vector divided by the square of the speed of light per unit volume. It is what radiation pressure pushes with, and without it momentum would not be conserved when charges act on each other through fields.

Named by 4 essays across one field — each of them below, with the objects they name alongside it.

Angular momentum that was in nothing at all. A ring carrying 10⁻⁶ C on a freely pivoted disc, with a solenoid on the axis threading 0.002 Wb through it, switched off over 1 second. Nothing is turning at the start and nothing has been touched. The collapsing flux drives a circumferential electric field round the ring, the ring is torqued, and the disc ends up spinning with 3.183·10⁻¹⁰ kg m²/s of angular momentum. Where was it? The two curves are the field's share, ε₀∫r × (E × B), and the matter's share integrated from the torque — computed by different routes and summing to a constant to 4.2e-15 of the total throughout. So the angular momentum was there before the switch was thrown, in a static electric field crossed with a static magnetic one, in a room where nothing whatever was moving. It is qΦ/2π, it does not depend on the radius of the ring or the shape of the solenoid, and it is the plainest demonstration available that the field is not a bookkeeping device for forces between distant charges.

The angular momentum that is in nothing at all

A charged ring and a solenoid, both at rest, with nothing moving anywhere. Switch the solenoid off and the ring starts to turn. Angular momentum is conserved, so it was there before the switch was thrown — and it was not in the matter, because nothing was moving. It was in the field, and it is qΦ over 2π whatever the geometry.

electromagnetism · Field energy The block has to move, and which way decides the momentum. A pulse of light crosses a free block of glass of index 1.5 and length L, with time in units of L/c. The pulse falls behind where it would have been in vacuum by 0.50 L while it is inside. The block's displacement, in units of EL/Mc², is obtained from momentum conservation alone: if the light inside carries Abraham's E/nc the block takes up the rest of E/c, moves forward, and ends 0.500 units along — Balazs's (n − 1). The sum of the pulse's lag and the block's shift, which is the centre of energy's departure from uniform motion, is then zero to within 0.0004 units at every moment. If instead the light carried Minkowski's nE/c, the block would have to recoil backwards and the centre of energy would end 1.25 units behind a straight line, which no isolated system can do. For a one-joule pulse through a kilogram of glass ten centimetres long the real shift is 5.6 × 10⁻¹⁹ m.

The momentum light carries into glass

A pulse of light in glass has two momenta, nE/c and E/nc, and for a century the argument was about which one is right. Both are measured. A free block that the light crosses must move forward, which only Abraham's value explains; a Cherenkov cone exists at all, which only Minkowski's allows. They answer different questions, and the difference between them is momentum carried by the glass.

electromagnetism · Field energy Where a moving charge keeps its field's energy and its momentum. Left: the density of field energy and of field momentum around a charged shell moving slowly to the right, by direction, each drawn as the mass per unit volume it contributes — energy density over c², momentum density over velocity. The energy density is the same in every direction; the momentum density goes as sin²θ, zero ahead and behind and largest broadside, because it is ε₀E × B and the magnetic field of a moving charge vanishes along its line of motion. Broadside it is twice the energy's; averaged over all directions sin²θ is 0.6667, and twice that is the 4/3. Right: the fraction of the field's energy, and of its momentum, lying within a distance r of the centre of a shell of radius a. Both are 1 − a/r, because both densities fall as the fourth power of distance: half of each lies within 2a and nine-tenths within 10a.

The mass a charge's field gets wrong by a third

Give a charged sphere's field its energy U and divide by c², and the field has a mass. Move the sphere and ask the field's momentum what the mass is, and it says four-thirds of that. The discrepancy took from 1881 to the 1960s to understand, and the answer is not a better calculation of the field. It is that a charge cannot hold itself together, and whatever does has energy too.

electromagnetism · Field energy Two pieces of current that one law makes push each other unequally. Two short elements of current a unit distance apart: element 1 pointing along the line joining them, element 2 at 45° to it. Left, the forces by the law that follows from Biot and Savart (Grassmann's): element 1 pushes nothing on element 2, because 1 points straight at it, while element 2 pushes on element 1 with a force of 0.71 units at right angles to it; the pair would feel a net 0.71 units. Right, the forces by Ampère's own law of 1826: 0.71 units each, along the line joining the elements, equal and opposite. Both laws give the same force between any two complete circuits, so no experiment on closed circuits can say which is right.

The force between pieces of a current

Ampère spent six years finding the force one short piece of current exerts on another, and the law he found is not the one used today. The modern law, which follows from the Biot–Savart field, disagrees with his for almost every pair of pieces — his obeys Newton's third law and the modern one does not. Both give exactly the same force between any two complete circuits, which is the only thing that has ever been measured. The force between two pieces of current is a quantity with two correct values, because a piece of current on its own does not exist.

electromagnetism · Ampere law

Named alongside it

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

Ampere lawAmperes force lawAngular momentumBiot savart lawCentre of massCherenkov radiationClassical electron radiusConservationCurrent elementDispersionElectromagnetic massField energy

All concepts