A long straight hollow conductor in air has inner radius and outer radius

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

Electricity, Magnetism and Electromagnetism, Ch Ampère's law, magnetic flux and the shape of the magnetic field

Problem

A long straight hollow conductor in air has inner radius

a = 4.0\,\mathrm{mm}

and outer radius

b = 8.0\,\mathrm{mm}

. It carries a steady current

I = 60\,\mathrm{A}

parallel to its axis, distributed uniformly over its cross-section. Treat the conductor as infinitely long, so that end effects are ignored. Let

s

be the perpendicular distance from the axis. ((a)) State the symmetry that fixes the shape of

\mathbf{B}

, and derive, using Ampère's law with circular loops, expressions for

B(s)

in the three regions

s < a

a \le s \le b

and

s > b

, in terms of

\mu_0

I

a

b

and

s

. State the direction of

\mathbf{B}

in words. ((b)) Evaluate

B

s = 6.0\,\mathrm{mm}

and at

s = 12\,\mathrm{mm}

. ((c)) A long air-cored solenoid, treated as infinitely long, carries the same steady current of

60\,\mathrm{A}

. How many turns per metre must it have for the field inside it to equal the field at the outer surface of the tube,

s = b