Is the coil the only viable shape for an electric engine ?

[zunf]

Hi, I'm an hobbyist
right now I'm into electric engines, simple engines for low voltage and low current applications; I have noticed that there are many variations about how and what an electric engine can do even considering only DC motors, but basically all the engines that I have examined up until now have something in common, and that is the presence of a coil, a copper coil, used as a conductor in the internals of the engine itself .

I wonder if such twisted copper cables are the only way to induce current in a DC / AC engine, for example can you replace a coil with a ring of copper ? A cylinder ?

What is the physical explanation that motivates the use of a coil and what makes it efficient compared to other shapes / solutions ?

After all the pulse of current is given by the change in the magnetic flux, which basically refers to the rotating parts of your engine, up until now I haven't found any explanation as to why it's an optimal thing to use a coil rather than a cylinder, or a sphere, or any other piece of geometry in metal form .
There are formulas that dictate the ins and outs of an optimal coil given the specs of an electrical engine ?

 

[meBigGuy]

I'm not a motor expert. But, check out these:
http://www.infolytica.com/en/applications/ex0113/
http://www.goldenmotor.com/frame22.htm

 

[Jeff Rosenbury]

There are lots of motor topologies. There are disk motors, linear motors, solid plate motors, etc.

The usual reason coils are used is that the flux increases with the square of the number of turns. This gives lots of flux for little current.

Linear motors are used in rail guns. Disk type motors are lightweight and cheap using traces on PCB boards as their conductors. The Faraday Disk was the first electric generator and used a solid plate.

A Van de Graaff generator doesn't even use metal in its moving parts. It relies on the electric field rather than the magnetic field.