Solenoids and Toroids

Picture a long solenoid as many circular loops stacked side by side. Each loop makes a weak field on its own, but packed close together their fields add up straight down the central axis while the sideways parts cancel. Outside the solenoid, the fields from neighboring loops point opposite ways and nearly wipe each other out. The result: a strong, uniform field inside, almost nothing outside, as long as the solenoid is long compared to its width.

To get the number, draw a rectangular Amperian loop: one side running inside the solenoid parallel to the axis, length L, the opposite side running outside where B is zero, and two short sides crossing the windings where B is perpendicular to the path and contributes nothing. Ampere's law says the loop integral of B dot dl equals mu0 times enclosed current. Only the inside side contributes, giving B L = mu0 (n L) I, where n is turns per length, so B = mu0 n I. Notice the solenoid's radius never enters: squeeze or stretch it sideways and B is unchanged, because the loop only cares about turns per unit length along the axis, not how far those turns reach outward.

A toroid is a solenoid bent into a donut, so its field lines close on themselves inside the core with nothing leaking out the ends. A circular Amperian loop of radius r drawn inside the core encloses all N turns, giving B(2 π r) = mu0 N I, so B = mu0 N I/(2 π r). Unlike the solenoid, this field depends on r, weaker farther from the center, which sets up the next idea: how changing these fields in time drives induced EMF.