A long, straight, solid cylinder, oriented with its axis in the z$-$

direction, carries a current whose current density is$$

J$.$ The current

density, although symmetric about the cylinder axis, is not constant

but varies according to the relationship

$$

J$=$

$2$Io

$$

a$0[1($r

a$)2]$

k for r$$ a

$$

J $=0$ for r$0$

where a is the radius of the cylinder, r is the radial distance from

the cylinder axis, and I$0$ is a constant having units of amperes.

$($a$)$ Show that I$0$ is the total current passing through the entire cross

section of the wire.

$($b$)$ Using Ampere$$s law, derive an expression for the magnitude of

the magnetic field$$

B in the region r$$ a$.$

$($c$)$ Obtain an expression for the current I contained in a circular

cross section of radius r$$ aand centered at the cylinder axis.

$($d$)$ Using Ampere$$s law, derive an expression for the magnitude of

the magnetic field$$

B in the region r $$ a How do your results in

parts $($b$)$ and $($d$)$ compare for r$=$ a$?$

Ans: $($b$)0$I

$2$r$,($c$)$I$0$r$2$

a$2(2$ r$2/$a$2),($d$)0$I$0$

$2$

r

a$2(2$ r$2/$a$2)$