A long, straight solid cylinder, oriented with its axis in the $z$ -direction, carries a current whose current density is $\overrightarrow{\boldsymbol{J}} .$ The current density, although symmetric about the cylinder axis, is not constant and varies according to the relationship
$$
\begin{aligned}
\overrightarrow{\boldsymbol{J}} &=\left(\frac{b}{r}\right) e^{(r-a) / \delta} \hat{\boldsymbol{k}} & \text { for } r \leq a \\
&=\mathbf{0} & \text { for } r \geq a
\end{aligned}
$$
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 $\overrightarrow{\boldsymbol{B}}$ 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 $\overrightarrow{\boldsymbol{B}}$ in the region $r \leq \underline{a}$. (e) Evaluate the magnitude of the magnetic field at $r=\delta, r=a,$ and $r=2 a$