$4-[30$ Marks Total $]$

a$-$A straight long solid cylindrical conductor of radius a carries current I along its length. The current density varies as $J=J_{0}r$ within the conductor, where $r$ is the radial distance from the axis of symmetry of the conductor.

$($a$)$ Find the constant $J_0$ in terms of quantities given

$($b$)$ ind the magnetic field strength for $r>ax-yz=0J(\frac{A}{m})yzdzr.$

$(c)$ Find the magnetic field strength for $r>a$

$b-An$ infinite sheet lying $in$ the $x-y$ plane $atz=0,$ has uniform current density $J(\frac{A}{m}),$ flowing $in$ the positive $y$ direction. Calculate the magnetic field $(magnetic$ flux density, $B)$ a distance $z$ above the sheet. Indicate field direction.

$c-$Consider two very long and very wide current sheets each carrying a uniform current density, $J0$ but $in$ opposite directions $as$ shown $in$ the figure. The sheets have a thickness $ofd$ and the current $is$ flowing uniformly through the thickness $of$ the sheets.

Calculate the magnetic field, $B,$ for all values $ofz.$