Magnetic energy in finite solenoids: I. The ideal current-sheet model

The infinite solenoid is one of the standard idealisations of introductory electromagnetism: its magnetic field is uniform inside the bore and zero outside. A finite solenoid is more instructive. Its fringing field stores a non-negligible amount of magnetic energy outside the bore, so the magnetic energy becomes a global quantity rather than a purely local one. We present a compact calculation, suitable for advanced undergraduate or beginning graduate electromagnetism, of the inside/outside energy partition of an ideal finite thin-walled solenoid. The central idea is to combine the exact total energy obtained from the known self-inductance with a direct numerical evaluation of only the interior energy. The exterior energy is then found by subtraction, avoiding any integration over an unbounded exterior domain. This finite-domain route gives a reproducible computational project and leads to simple analytic checks. In the short-solenoid limit the exterior energy dominates only logarithmically, while in the long-solenoid limit the absolute exterior energy approaches a finite end-field constant and the leakage ratio decreases inversely with aspect ratio. We also give a three-parameter pocket formula for the exterior-to-interior energy ratio, accurate to about one per cent over the range normally useful for classroom estimates.