Problem $3$

Magnetic levitation can be achieved by utilising magnetic pressure, which arises from strong magnetic fields generated by current$-$carrying coils, to create lift. From this week lecture, we have learnt that a uniform current density may give rise to a vector magnetic potential as below

$A=-$hat$(z)\frac{{}_0{J}_0(x^2+y^2)}{4},(W\frac{b}{m})$

$($a$)$ Use the given vector magnetic potential to find magnetic field vector $H.$

$($b$)$ Apply the vector Poisson's equation to find the current density $J.$ Will your result hold for time$-$vary fields? Explain the reason to your tutor.

$($c$)$ In tandem with Ampere's law, find magnetic field vector $H.$ Have you observed any difference compared with the result obtained in part a of this problem?