A long straight wire lies along the axis and carries a steady

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 lies along the

z

axis and carries a steady, uniform current density of magnitude

J = 1.5\times10^{6}\ \mathrm{A\,m^{-2}}

in the

+z

direction. Inside the wire the magnetic field is

\mathbf{B} = \tfrac12\mu_0 J\left(-y\,\hat{\mathbf{x}} + x\,\hat{\mathbf{y}}\right),

with

x

and

y

in metres and

\mu_0 = 4\pi\times10^{-7}\ \mathrm{T\,m\,A^{-1}}

. Take this expression to hold at every interior point. ((a)) Compute

\nabla\cdot\mathbf{B}

inside the wire. ((b)) Compute

\nabla\times\mathbf{B}

inside the wire, giving its magnitude and direction.

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