Alternating-current properties of a copper conductor clad in a magnetic sheath

Formal expressions are derived for the resistance to alternating current of a copper conductor clad in a magnetic sheath and for the magnetic field within the sheath. A method of numerical evaluation is outlined, and curves, calculated for a frequency of 2250 c/s, are given for a copper wire of radius 0.2 mm clad in a nickel sheath up to 0.4 mm thick. Two values of sheath permeability are used: one high, corresponding to currents below 250 mA, and the other low, corresponding to currents of 5 A or more. The assumption of constant permeability made in the theory is discussed in relation to the known properties of nickel, and the limitations of the calculations are outlined. The possibility of a significantly non-linear current/voltage relationship is noted. Limiting forms of the solution are given for frequencies greater than 1 Mc/s, and it is shown that, while a thin protective coating of nickel is unlikely to have any significant effect in the audio-frequency range, it could completely dominate the behaviour of the conductor at radio frequencies.