A solenoid is a magnetic energy storage device made by winding wire over a (hollow or filled) cylinder. A toroid is any doughnut-shaped object (i.e. a cylinder bent into a circle with its ends joined). Usually, the windings of a solenoid have a circular cross-section, but the figure below depicts a toroidal solenoid of rectangular cross-section. Fewer than half of the windings are drawn to reduce clutter. We choose the rectangle because it makes the integration easier. Current of amps flows through the wire, creating a magnetic field inside the toroid given by the equation in the figure. Here, N is the total number of turns of wire and r is given in the figure. The direction of the field is given by the right-hand rule.
Starting with Φ = ∫B·dA, determine the equation for the magnetic flux through the cross-sectional area A: Φ_total = N·B·A.