A long straight wire of radius lies along the axis and carries a steady current…

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Electricity, Magnetism & Electromagnetism · Vector calculus and field geometry

Electricity, Magnetism and Electromagnetism, Ch Divergence, curl and the local form of the field laws

Problem

A long straight wire of radius

a = 1.5\ \text{mm}

lies along the

z

axis and carries a steady current

I = 6.0\ \text{A}

spread uniformly over its cross-section. A student wants the local form of Ampère's law and writes: "Take a circular loop of radius

s = 5.0\ \text{mm}

round the wire, traversed so that the normal of the flat disc it bounds is along

+\hat{\mathbf{z}}

. Ampère's law gives

\oint\mathbf{B}\cdot d\mathbf{l} = \mu_0 I

. By Stokes's theorem the left side is

\int(\nabla\times\mathbf{B})\cdot d\mathbf{A}

over the disc, whose area is

\pi s^2

. Therefore

(\nabla\times\mathbf{B})_z = \frac{\mu_0 I}{\pi s^2} = 9.6\times10^{-2}\ \text{T m}^{-1}

at every point of the disc, including points in the empty space outside the wire." ((a)) Identify the faulty step and say what should have been written in place of

\mu_0 I

on the right-hand side. ((b)) Give the correct value of

(\nabla\times\mathbf{B})_z

at a point

5.0\ \text{mm}

from the axis. ((c)) Give the correct value of

(\nabla\times\mathbf{B})_z

at a point

1.0\ \text{mm}

from the axis, inside the wire.

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