ain properties and demonstrate how the $$grad$,$div$$ and $$curl$$

operators may be expressed in terms of the del operator.

$[5$ marks$]$

$($ii$)$ Explain briefly the concepts of tubes and slices as applied to

the flow of steady current through a resistive material. Clarify how

dual bounds of resistance may be estimated and provide some

physical explanation of why such bounds occur.

$[6$ marks$]$

$($iii$)$ Define the $$divergence$$ of a vector field mathematically and

provide a physical insight. Explain why an electric field, D$,$ can

have non$-$zero divergence, but a magnetic field, B$,$ always has

zero divergence.

$[5$ marks$]$

$($iv$)$ In a region of free space the electrostatic potential V is given

by V $=$ axy$1$

$$bxz$.$ Derive the expression for the electric flux

density D$,$ calculate div D and comment on the distribution of free

electric charges in the region.

$[6$ marks$]$

$($v$)$ Starting from the fundamental equations related to relevant

field vectors, show that an electrostatic field may be described by

Poisson$$s equation in terms of the electrostatic potential

$2$

V

$=$

$$

o

$[6$ marks$]$

$($vi$)$ Define the terms permeability, permeability of free space and

relative permeability. Comment on the reason and appropriateness

of the often made assumption that the permeability has a constant

value and explain how this relates to the properties of practical

magnetic materials.