Question

18. A infinitely long cable consists of a solid cylindrical inner conductor of radius a, surrounded by a concentric cylindrical conducting shell of inner radius b and outer radius c. The inner conductor has a non-uniform current density (r) = ar in the z direction shown. a is a positive constant with units A-m'. The outer conductor has a uniform current density: Jr) = -B (in negative z). B has the same unit as a. The conductors carry equal and opposite currents of magnitude lo Joute total current lo total current lo Z dir A. Find expressions for a and B in terms of a, b, c, and lo B. Determine the magnitude and direction of the magnetic field for the regions (i) r<a, (ii) a <r<b, (iii) b<r<c, (iv) and r>c. For each region, redraw the coaxial cable clearly indicating your choice of Amperian loop and associated parameters.

Answer 1

Problem 28 A i no he inner conductor has a current density J(1) =ar and the outer conductor has a current density J) =-p . A) so, Io = $ 10) ds za friz trde = 294x ? 2 ITO T2 TT SO 2 TTT Gausian Surface similarly, Io --PJ” ardo --2012-10-22 «, pa neCAP 1 B) ;) Magnetic field at area is given by : $8.- Mo Sg.ds , B(ra) - SarZárdr , B = 0 ar? Como corza 5 1 In the region aarab , the magnetic field is, B=0 as there is no current density. iii) an the region bar<e the magnetic field is , B=0 as also there is no current- i density exista i în Bul- for region rye the magnetic field is given by , & Bick = Mo J J. ds or, B.27r - Mo Io or, B(r>c) , hobo (r>c) no curren son arsel Grau Amberim 2 TT loops or, B = Moßer for rac