Magnetic Field of a Solenoid and a Toroid

Having established Ampere's law and demonstrated its power on the simplest possible case (a single long straight wire, Section 10.15), this section turns to two more elaborate, but practically very important, current geometries: the SOLENOID -- a long, tightly and closely wound helical coil of wire -- and the TOROID -- essentially a solenoid bent around into a closed ring or tube shape. Both devices are, in effect, ways of concentrating and shaping a magnetic field using many turns of current-carrying wire, and both are analysed, in the two subsections that follow, using exactly the Ampere's-law strategy just demonstrated: choose an Amperian loop that matches the symmetry of the coil's winding, so the line integral $\oint\vec{B}\cdot d\vec{l}$ can be evaluated …