Solved: A long, straight, solid cylinder, oriented with

Chapter 28, Problem 28.71

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A long, straight, solid cylinder, oriented with its axis in the z-direction, carries a current whose current density is J S . The current density, although symmetric about the cylinder axis, is not constant but varies according to the relationship J S = 2I0 pa2 c 1 - a r a b 2 d k n for r a = 0 for r a 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 Amperes law, derive an expression for the magnitude of the magnetic field B S 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 Amperes law, derive an expression for the magnitude of the magnetic field B S in the region r a. How do your results in parts (b) and (d) compare for r = a?