CALC A long, straight, solid cylinder, oriented with its axis in the z-direction, ca...

CALC A long, straight, solid cylinder, oriented with its axis in the z-direction, carries a current whose current density is The current density, although symmetric about the cylinder axis, is not constant but varies according to the relationship

where a is the radius of the cylinder, r is the radial distance from the cylinder axis, and I₀ is a constant having units of amperes. (a) Show that I₀ 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 in the region r ≥ a. (c) Obtain an expression for the current I contained in a circular cross section of radius r ≤ a and centered at the cylinder axis. (d) Using Ampere’s law, derive an expression for the magnitude of the magnetic field in the region r ≤ a. How do your results in parts (b) and (d) compare for r = a?