Along, straight, solid cylinder, oriented with its axis in the z-direction, carries a current whose current density is $\overrightarrow{J}$ . The current density, although symmetric about the cylinder axis, is not constant and varies according to the relationship

$$\overrightarrow{J}=(\begin{array}{c}\frac{b}{r}\end{array}){e}^{(r-a)/\delta}\hat{k}\text{}for\text{}r\text{}\le a$$

$=0f{\textstyle \phantom{\rule{1em}{0ex}}}\text{or}{\textstyle \phantom{\rule{1em}{0ex}}}\ge a$

where the radius of the cylinder is a = 5.00 cm, ris the radial distance from the cylinder axis, b is a constant equal to 600 A/m, and $\delta $ is a constant equal to 2.50 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, d, 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 $\overrightarrow{B}$ in the region $r\ge a$. Express your answer in terms of $I}_{0$ rather than b.

(c) Obtain an expression for the current | contained in a circular cross section of radius r... 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 $\overrightarrow{B}$ in the region $r\ge a.$

(e) Evaluate the magnitude of the magnetic field at r- = 6, r= a, and r= 2a.