An ideal solenoid of radius a has n turns per unit length and current I. The magnetic flux ΦB through any circular area of radius a inside the solenoid, centered on and perpendicular to the solenoid axis, is

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The magnetic field inside a solenoid is given by B = μ0nI, where μ0 is the permeability of free space, n is the number of turns per unit length, and I is the current. Since the magnetic field is uniform over any circular area inside the solenoid, the magnetic flux through the area is simply the product of the magnetic field and the area, ΦB = BA. Since the circular area has radius a, its area is A = πa^2. Therefore, ΦB = BA = μ0nIπa^2.

An Ideal Solenoid of Radius a Has N Turns Per