Illustrated Physics
physics, drawn
Concept

Equipotential — where it appears

A surface on which the potential is constant, met at right angles by every field line and free to be replaced by a conductor. That last freedom is a calculating tool: any equipotential of a known problem can be made into a metal surface without disturbing the field outside it.

Named by 6 essays across 2 fields — each of them below, with the objects they name alongside it.

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.

One number for every point, and nothing at all is lost

The electric field is three numbers at every point of space. Replacing it with one number loses nothing — and the reason it loses nothing is the same reason a hill can be drawn as a contour map.

electromagnetism · Potential A conductor in a field, with the surface charge solved for. Field lines approaching an isolated conducting cylinder. The surface charge was found by requiring the conductor to be an equipotential, and the lines then end on that charge, meeting the surface at right angles and leaving the interior empty.

The inside of a conductor, where the field is exactly nothing

Put a metal object in any electric field and the field inside it is zero. Not small — zero, by an argument that takes one sentence, and with consequences that reach from lightning to the most precise test of Coulomb's law ever made.

electromagnetism · Conductors Two plates 0.20 plate-widths apart, with the field traced. The electric field between two oppositely charged plates separated by 0.20 of their own width, traced by following the field of 26 discrete charges on each plate rather than drawn as parallel lines. In the middle the lines are straight and evenly spaced; near the ends they bow outward. The field nine-tenths of the way to the edge is 80 per cent of the field at the centre.

How much charge a shape will hold, before anything is charged

Capacitance is decided by geometry alone. Two pieces of metal have a number attached to them, fixed by their shape and their separation, and it is settled before any charge arrives.

electromagnetism · Conductors 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 The surface a spin decides. The free surface of a liquid in a dish of radius 0.5 m turning at 10, 20, 40 revolutions a minute. In the rotating frame the surface is a level set of gz − ½ω²r², so it is a paraboloid exactly and not to some approximation, and nothing about the liquid appears in its shape: the same curve is got with mercury, water or oil. The rim stands 14.0 mm, 55.9 mm, 223.6 mm above the centre at those rates. A parabola z = r²/4f has focal length f, so these surfaces are mirrors of focal length g/2ω² — 4.47 m at 10 rpm, 1.12 m at 20 rpm, 0.28 m at 40 rpm. That is checked here on the drawn curves rather than quoted: a vertical ray reflected off the surface at a quarter, a half, three quarters and the whole of the radius crosses the axis at the same height to 0.00%, which is what a mirror with no spherical aberration means. Doubling the spin quarters the focal length, and there is no other adjustment: the dish can only ever look straight up.

The surface a spin decides

Spin a dish of liquid and its surface settles into a paraboloid — exactly, with nothing about the liquid in the shape. A parabola of that form has a focal length of g over twice the spin rate squared, so a bucket of mercury turning at twenty revolutions a minute is a telescope mirror figured by a clock instead of by grinding.

fluids · Hydrostatics The current crowding into a contact spot. A meridional section through a circular contact between two solids, with the spot at the centre. The closed curves are equipotentials and the curves running through the spot are current lines, each carrying an equal share. Both are exact: in the coordinates built on the spot's rim the equipotentials are confocal spheroids and the current lines confocal hyperboloids, and in this section they are ellipses and hyperbolas. What the picture shows is that the current has to converge from a region many spot radii across and then spread again, and that the potential falls almost entirely within a few radii of the contact. The equipotential drawn at 12% of the drop sits 5.2 radii away, and everything beyond it contributes that last 12%.

The resistance that is a length

Two metals touching do not touch over the area they appear to. Current crosses at a few small spots and has to converge into each one, and the resistance of that convergence contains no area and no path length — only the size of the spot, divided into the resistivity.

electromagnetism · Conductors

Named alongside it

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

Boundary conditionsElectric fieldConductorSurface chargeElectric potentialElectrostatic shieldingApparent weightCapacitanceCentrifugal forceConductivityConductorsConic section

All concepts