Illustrated Physics
physics, drawn
Concept

Antenna — where it appears

A structure that converts a current into a radiated field or the reverse, with a directional pattern set by its size in wavelengths. Reciprocity makes the two patterns identical, so an antenna listens exactly as it speaks and only one of the two need ever be measured.

Named by 4 essays across 3 fields — each of them below, with the objects they name alongside it.

Two loops, two calculations, one number. The coupling between a circular loop of radius 10 cm and a rectangle of 5 by 3 cm tilted at 35° and offset sideways, against how far apart they are along the axis. The two loops differ in area by a factor of 21 and in shape entirely. Two quantities are plotted and they lie on top of one another. One is the flux through the rectangle when a current runs in the circle, obtained by integrating the circle's Biot–Savart field over the rectangle's tilted surface. The other is the flux through the circle's disc when the same current runs in the rectangle, integrated over a disc a hundred times the area. Nothing is shared between the two calculations except the positions of the wires, and they agree to 0.075 per cent — a difference which is the quadrature's, not the physics's. This is reciprocity, and it is not obvious: there is no reason from the geometry why a small loop should catch as much of a big one's field as the big one catches of the small one's. It follows from the double line integral the two quantities can both be reduced to, which is symmetric in the two loops — and that reduction is the argument, not this figure, which is the check that the argument is true of the actual fields.

The coupling that is the same both ways

A small loop and a large one catch the same fraction of each other's field. Nothing in the geometry suggests it — one has twenty-one times the area of the other — and the two quantities are computed here by two integrals with nothing in common, over two surfaces of different shapes, agreeing to seven parts in ten thousand.

electromagnetism · Induction Nine slabs, no symmetry, and one transmission. A stack of 9 slabs of random wavenumber and random thickness, with no symmetry anywhere in it. Send a wave in from the left and the amplitude that emerges on the right is 0.6694240937; turn the stack round and send it in from the right and the amplitude is 0.6694240937. The two agree to every digit the arithmetic has. This is not a property of the stack — it survives any arrangement, any number of layers, any amount of internal reflection — but of the equation, which is unchanged when the sign of time is reversed, and the phase is protected as well as the modulus, which is the stronger statement and the one an interferometer would notice. The reflections are a different matter. Their moduli are equal too, at 0.742880, but only because nothing here absorbs; their phases differ by 1.621 radians, because the wave meets a different first surface from each side. What the theorem protects is the pair of ends, and not what the wave does on its way between them.

Swap the ends and nothing changes

Put a source at one end of the most complicated arrangement of materials anybody can build and a detector at the other, then exchange them. The reading is identical — not approximately, not on average, but to every digit the arithmetic has. Two things in physics break it, and neither of them is a shape.

waves · Wave motion A parallel beam brought to a point on the far side. Parallel rays sent into Luneburg's sphere, whose index falls from the square root of two at the centre to one at the rim, from the left. Each beam is brought to a single point on the rim diametrically opposite the direction it came from; the worst of the 11 rays misses by 2.0e-3 radii. The sphere has no axis — every direction is the same to it — so a beam from any direction is focused as well as any other, and there is no such thing as an off-axis aberration.

The lens with no axis

Every ordinary lens has an axis, and every one of its hardest aberrations is a penalty for looking away from it. A sphere has no axis, so a sphere cannot have off-axis aberrations — but a glass ball has spherical aberration instead, because one index and one curvature are too few to focus every ray. Luneburg's sphere, whose index falls from √2 at the centre to 1 at the rim, keeps the symmetry and removes the aberration: it brings parallel light from every direction to a perfect point on its own surface.

optics · Fermat A field that oscillates faster than any of its components. The oscillation of (cos x + i·2·sin x) to the power 10 over half a period, with its changing size divided out so that only the turning of its phase shows, beside the fastest wave it contains, cos 10x (dashed). Multiplied out, the function is a sum of waves e^(inx) with n running from −10 to 10 and no further. Yet in the shaded window, |x| < 0.62, it oscillates faster than cos 10x — near the origin like cos 20x, 2 times faster — completing 3.0 cycles there where its fastest component completes 2.0. Outside the window it slows down, and the next figure shows what happens to its size there.

The wiggle faster than any wave in it

Light passing through a lens carries no detail finer than half a wavelength, because no travelling wave in it varies faster than that. It would seem to follow that the light itself cannot vary faster anywhere. It can. A sum of slow waves can be arranged to wiggle faster than any of them over a stretch of any length chosen — and to focus to a spot as narrow as wanted — provided the stretch sits between side lobes that grow exponentially with every extra wiggle.

waves · Huygens

Named alongside it

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

LinearityReciprocityBandwidthBiot–SavartDetectorDiffraction limitEvanescent waveFaraday's lawFermat's principleFocal lengthFourier transformGraded-index

All concepts