A long coaxial cable has an inner conductor of radius carrying a steady current along

PHYSICS · Multi-part · Fermi University Physics · Electricity, Magnetism & Electromagnetism · Magnetostatics · Magnetostatics

The Boundary-Value Problem, Ch The vector potential and the magnetostatic boundary-value problem

Problem

A long coaxial cable has an inner conductor of radius

a = 1.0\,\mathrm{mm}

carrying a steady current

I = 3.0\,\mathrm{A}

along

+\hat{\mathbf{z}}

, and a thin outer conducting shell of radius

b = 5.0\,\mathrm{mm}

carrying the return current

I

along

-\hat{\mathbf{z}}

. Between the conductors (

a < s < b

) the field is

\mathbf{B} = \dfrac{\mu_0 I}{2\pi s}\hat{\boldsymbol{\phi}}

, and outside the shell (

s > b

)

\mathbf{B} = 0

. Take

\mathbf{A} = A_z(s)\hat{\mathbf{z}}

and fix the arbitrary constant by setting

A_z = 0

s = b

, consistent with

\mathbf{A}

being constant (and continuous) in the field-free region beyond the shell. ((a)) Find

A_z(s)

for

a \le s \le b

. ((b)) Evaluate

A_z

s = 2.0\,\mathrm{mm}