A thick-walled copper pipe (inner radius a and outer radius b) is centered on the z-axis and carries a current density of magnitude J(r)=βr. (The parameter β has units A/m³ .) A wire runs parallel to the pipe a distance R=2b away from the z-axis.

(a) What is B=|B^→|, the magnetic field magnitude, inside the pipe (r < a), inside the copper itself (a < r < b), and outside the pipe (r > b)?

(b) Sketch a plot of the field magnitude B(r). Be sure to clearly label the axes of your plot, the value of the maximum B_max, the inner and outer radii a and b, and how B(r) depends on r in each region described in part (a).

(c) Suppose that a current i_ω flows in the wire at R=2b. What force does the wire exert on the pipe if i_ω flows in the same direction as i_p, the total current in the pipe?