A long, straight, solid cylinder, oriented with its axis in the $z$ -direction, carries a current whose current density is $\vec{J}$ . The current density, although symmetric about the cylinder axis, is not

constant and varies according to the relationship $$ \begin{aligned} \vec{J} &=\left(\frac{b}{r}\right) e^{(r-a) / \delta} \hat{k} & \text { for } r \leq a \\ &=0 \quad \quad \text { for } r \geq a \end{aligned}

$$ where the radius of the cylinder is $a=5.00 \mathrm{cm}, r$ is the radial distance from the cylinder axis, $b$ is a constant equal to $600 \mathrm{A} / \mathrm{m},$ and $\delta$ is a constant equal to 2.50 $\mathrm{cm} .$ (a) Let $I_{0}$ be the total current passing through the entire cross section of the wire. Obtain an expression for $I_{0}$ in terms of $b, \delta,$ and $a$ . Evaluate your expression to obtain a numerical value for $I_{0}$ . (b) Using Ampere's law, derive an expression for the magnetic field $\vec{\boldsymbol{B}}$ in the region $r \geq a .$ Express your answer in terms of $I_{0}$ rather than $b$ . (c) Obtain an expression for the current $I$ contained in a circular cross section of radius $r \leq a$ and centered at the cylinder axis. Express your answer in terms of $I_{0}$

rather than $b .$ (d) Using Ampere's law, derive an expression for the magnetic field $\vec{\boldsymbol{B}}$ in the region $r \leq a .$ (e) Evaluate the magnitude of the magnetic field at $r=\delta, r=a,$ and $r=2 a .$