This universe pins the force between charges at 1/r in four dimensions of space, where Gauss's law would give 1/r² and no atoms (why). Here's what that does to electric and magnetic fields, as things to play with. In short: point charges can't tell, and everything else can. As across this site, 3D is warm and 4D is ice.
Two charges, seen in a flat slice through them, coloured by the potential (warm positive, ice negative; the lines are equipotentials). Drag them. Then lift the positive one out of the slice, into the fourth direction, w (in 3D, into the third). Switch laws to compare.
That's the pin's whole point. Between point charges the potential is 1/distance, and distance is distance in any number of dimensions. Lifting a charge into w does exactly what lifting it into z does for us. Gauss's 1/r² is a different law: steeper close in, gone sooner.
Draw a charge's field as lines, as many crossing a sphere as the field's flux through it. Under Gauss's law, in any dimension, the flux is the charge, whatever the sphere's size: every line starts on a charge. The pinned field falls as 1/r², but in 4D the sphere around a charge grows as r³, so the flux grows in proportion to the radius. New lines have to begin in empty space.
ours, or 4D Gauss: lines only from the charge
4D pinned: new lines start everywhere
It's not a paradox: in the vacuum this universe chose (the lens vacuum), the field D that Gauss's law counts still starts and ends only on charges. The field E that pushes charges is D seen through a lens, and the lens makes empty space around a charge act as if it held a faint charge of its own sign, falling as 1/r³. Summed out to radius r, that grows as r. The full working.
Where 4D parts from ours is spread-out charge. Make an atom as a nucleus (+1) inside a thin shell of electron charge (−1) of radius a, and feel it from outside with a test charge. In our world Newton's shell theorem says a shell looks like a point at its centre, so the two cancel and the atom is invisible. In 4D under the pin, the shell looks a little smaller than a point would. So the atom is faintly positive from outside: it pulls electrons and pushes protons, as a²/8s³. Inside, a lone shell's potential isn't flat: no Faraday cage.
This is the root of most of what's different about 4D chemistry. Two neutral atoms attract permanently, as R⁻⁵ (the forces chapter). Noble gases are held by the crowd and condense thousands of kelvin hotter than their wells suggest (gases). In a salt, an ion's cloud leaks a field its neighbours feel as R⁻³.
A magnetic field is a turning: it bends a moving charge's path within a plane. In 3D there are three planes, so three numbers, which we write as a vector. In 4D there are six planes (xy, xz, xw, yz, yw, zw), so B has six components. It's a field of little planes, not of arrows. Currents make it through the same lens as charges make E, so a current's field comes from the same 1/r kernel. B still has no sources (no magnetic monopoles). But Ampère's law fails, as Gauss's does.
Here is a ring of current's field, along a line through its centre at right angles to the ring, computed from the loop by Biot and Savart's sum with each law's kernel:
Under the pin, magnets reach as far as ours: a magnet's field falls as 1/r³, where plain 4D Maxwell would give 1/r⁴. Two oddities. In 4D a loop can't be threaded: a loop only links a surface, never a line or another loop, so no ring of current encircles a wire. And a straight wire's field falls as 1/ρ, like ours, though the space around a 4D wire is three-dimensional.
In a uniform field a charge's velocity turns. In our world it circles in one plane and drifts freely along the field. In 4D a field can turn two planes at once, xy and zw, at two rates. Set the two parts. The left panel shows the path's shadow on the xy plane, a plain circle. The right shows its shadow on xz, which mixes the two turnings.
shadow on xy
shadow on xz: both turnings at once
The two parts can also be read another way: as a left half and a right half (the field plus or minus its mirror through the other plane), the same split that divides 4D rotations. An electron's spin is handed: it feels only the left half (motion and spin). Make the two parts equal and the field is all left: the path closes, and the electron's spin feels the field fully. Make them opposite and it's all right. The path still bends, but the electron's spin doesn't feel that magnet at all.
The working: the force (why 1/r, the lens vacuum, the shell) · motion and spin (rotations in planes, handed spin) · where we are · four