Inside a shaped copper block a steady current flows with the current density J = (5.0…

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

Inside a shaped copper block a steady current flows with the current density

\mathbf{J} = \left(5.0\times10^{7}\,x\right)\hat{\mathbf{x}} - \left(2.0\times10^{7}\,y\right)\hat{\mathbf{y}} - \left(3.0\times10^{7}\,z\right)\hat{\mathbf{z}},

where

x

y

and

z

are in metres, the coefficients are in

\mathrm{A\,m^{-3}}

and

\mathbf{J}

comes out in

\mathrm{A\,m^{-2}}

. ((a)) Compute

\nabla\cdot\mathbf{J}

. ((b)) Compute

\nabla\times\mathbf{J}

. ((c)) Find the

x

component of

\mathbf{J}

at the point

x = 2.0\ \mathrm{mm}

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