Chapter 28, Problem 28.75P

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

$\begin{array}{cccc}\stackrel{\to }{\text{J}}& =& \left(\frac{\text{b}}{\text{r}}\right){{\displaystyle \text{e}}}^{\left(\text{r-a}\right)/\delta }\hat{\text{k}}& for\text{ r }\le \text{ a}\\ & =& 0& for\text{ r }\ge \text{ a}\end{array}$

where the radius of the cylinder is a = 5.00 cm, r is the radial distance from the cylinder axis, b is a constant equal to 600 A/m, and δ is a constant equal to 2.50 cm. (a) Let I₀ be the total current passing through the entire cross section of the wire. Obtain an expression for I₀ in terms of b, δ, and a. Evaluate your expression to obtain a numerical value for I₀. (b) Using Ampere's law. derive an expression for the magnetic field $\stackrel{\to }{\text{B}}$ in the region r ≥ a. Express your answer in terms of I₀ rather than b. (c) Obtain an expression for the current I contained in a circular cross section of radius r ≤ a and centered at the cylinder axis. Express your answer in terms of I₀ rather than b. (d) Using Ampere's law, derive an expression for the magnetic field $\stackrel{\to }{\text{B}}$ in the region r ≤ a. (e) Evaluate the magnitude of the magnetic field at r = δ, r = a. and r = 2a.