Magnetic energy in finite solenoids: II. Winding pitch and near-field leakage

Abstract The textbook solenoid is usually represented by a continuous azimuthal current sheet, but a real solenoid is wound turn by turn, introducing a length scale absent from the ideal model. We present a compact Fourier-Bessel treatment of this pitch-scale correction, intended for advanced undergraduate or beginning graduate electromagnetism. The winding is replaced by an axisymmetric surface current whose axial dependence is periodic. Each non-zero axial Fourier harmonic is solved exactly in terms of modified Bessel functions, and its magnetic energy is split into interior and exterior parts by a surface-energy identity. The calculation shows that short-wavelength winding-density modulation stores nearly equal energy on the two sides of the winding surface. For a long tightly wound solenoid this produces an extensive correction to both the interior and exterior energies and hence a pitch-controlled leakage floor. The same solution also shows that this extra exterior energy is a near-field effect: beyond a radial clearance of order the pitch it is exponentially suppressed. The model gives a compact setting in which Fourier series, cylindrical boundary conditions, modified Bessel functions and magnetic-field energy can be studied in one problem.