A long rod, radius R, that has uniform positive charge density p is spun around its central axis withangular velocity w, as shown below, so that the charges follow circular paths centered on the axis.Current is the flow of charge, so we have rings of current of varying radius r.You may want to review the following relationships:Current density J = dI/dA = pv (from chapter 24!) and v=wr (from PHY 204!)This means that the rotating rod of charge is equivalent to a non-uniform current density J = pwr,where the current is circulating around the axis and r is the radial distance from the axis. Note thishas the same symmetry as the solenoid (current flowing in circles, field lines straight and parallel tothe axis). The difference is that this is like having many nested solenoids, and as we head out fromthe axis, we see more and more current.RI will walk you through the steps for using Ampere's law to find the magnetic field inside the rod.(a) Draw a diagram of a cross section of the rod, cut across the diameter, and the appropriate testloop to use in applying Ampere's law in finding the magnetic field at a distance r from the axis.Be very careful with your diagram: the test loop is just like the one for the solenoid, so be honestabout what part of our source is inside the test loop!(b) Use part (a) to find an expression for how much current is enclosed in (pokes through) the testloop (in terms of p, w, r, R). Be sure to reference your diagram, and in particular note what partof cylinder is inside the test loop! Note also that since the current density is not uniform, youDO need to integrate. It will help to draw a slice of the loop with current dI and label it withthe distances r (the test point location, and location of one edge of the test loop,) R (surface ofthe rod,) and y (the distance from the axis to the slice of current.)(c) You now have the current enclosed by the test loop for the right hand side of Ampere's law.The other side (field times length) of the law is the same as in the original Ampere's law for asolenoid. Solve Ampere's law for the field as a function of r.(d) The field should be strongest at points on the axis (at r = 0) and decrease as we move away,going to zero when r = R. Why (physics argument) is this true? Check (math) that this isconsistent with the implications of your answer to part (d). If it is not, try to find your mistake!

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Answered: A long rod, radius R, that has uniform…

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