A model vector field

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Vector calculus and field geometry · Vector calculus and field geometry

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

Problem

A model vector field, offered as a candidate electric field, is

\mathbf{E} = k\left[(x + y)\,\hat{\mathbf{x}} + (y - x)\,\hat{\mathbf{y}}\right],\qquad k = 200\ \mathrm{V\,m^{-2}},

with

x

and

y

in metres and no

z

component anywhere. ((a)) Compute

\nabla\cdot\mathbf{E}

. ((b)) Compute the

z

component of

\nabla\times\mathbf{E}

, and state the sense in which a paddle wheel with its axis along

z

would turn, seen from the

+z

side.

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