7. (20 points) Coaxial Cable, A wire with radius Rj as shown in the figure carri

Question

7. (20 points) Coaxial Cable, A wire with radius Rj as shown in the figure carries a non-uniform current density/ = Zr flowing out of the page. The total current in the wire is I (you will determine the total current below). A cylindrical conducting tube with inner radius R2 and outer radius Ra surrounds the wire and has the same total current flowing into the page. R, (a) What are the units of Z? rn at is the magnetic field (as a function of an arbitrary parameter r and constants) inside the WIRE( Ri)?

Explanation / Answer

a] Unit of Z = unit of J/ unit of r = A/m^3

b] B*2pi r = u0* ienclosed

B*2pi r = u0* integral Zr 2pi r dr = u0* 2pi*Z*r^3/3

B = u0*Z*r^2/3

b-ii] B*2pi r = u0* integral Zr 2pi r dr = u0* 2pi*Z*R1^3/3

B =  u0*Z*R1^3/3r

c]In the graph, B will increase (parabolically) from zero at r=0 to  u0*Z*R1^2/3 at r=R1 and then decreases to u0*Z*R1^3/3R2 at R2

d] B*2pi r = u0* i enclosed and net current enclosed is zero

B = 0

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