Physics Friday 31

Suppose we have a coaxial cable, with an inner wire of radius a, and an outer metal sheath of inner radius b. A current I flows down the wire and back through the sheath. What is the self-inductance per unit length Λ of the cable?


In the space between the inner wire and the sheath, where changes in flux will produce a back-emf, only the current in the inner wire contributes to the magnetic field. Consider a rectangular area parallel to the cable axis, extending radially between the two conductors. This rectangle will have width r=ba; let us call it’s length s. The flux through the area s dr of an infinitesimal segment (in the radial direction) is B(r)s dr, as the field is perpendicular to the area. The magnetic field due to a long thin wire is , and thus the total flux through our rectangle is:
.
The flux per unit length is thus
,
and so the self-inductance per unit length is
.

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