Question

Please help with this electrodynamics problem. Jackson Electrodynamics Problem 5.35. Thank you very much.

5.35 An insulated coil is wound on the surface of a sphere of radius a in such a way as to produce a uniform magnetic induction B, in the z direction inside the sphere and dipole field outside the sphere. The medium inside and outside the sphere has a uniform conductivity o and permeability M. (a) Find the necessary surface current density K and show that the vector poten- tial describing the magnetic field has only an azimuthal component, given by A= Boa" "sin e (b) where r. (r) is the smaller (larger) of r and a. At t=0 the current in the coil is cut off. [The coil's presence may be ignored from now on. With the neglect of Maxwell's displacement current, the decay of the magnetic field is described by the diffusion equation, (5.160). Using a Laplace transform and a spherical Bessel function expansion (3.113), show that the vector potential at times : >0 is given by A. = 3Bya sin e [ e-kj. (6)(69) dk where y = 1/uoa is a characteristic decay rate and 1(x) is the spherical Bessel function of order one. Show that the magnetic field at the center of the sphere can be written explicitly in terms of the error function (x) as B.0, 1) = B.["(vm) - Vauwexp(-1)] (c) Show that the total magnetic energy at time 1 > 0 can be written W = DOC e-Zukti(k)] dk Show that at long times (vt >> 1) the magnetic energy decays asymptotically V2B ? 24p( vt) 32

Answer 1

Given An Insulated coil is wound on the surface of a sphere of rodius a in such a ways as to produce a uniforem magnetic Induction Bo in the 2- direction inside the sphere and dipole fixed outside the sphere the medium inside & outside the sphere has a un form conductivity & & permeability M e-dipole outside vinside We have to determine the surface current density K and have to prove that Ad- Boarra sind VA We know that Н= Mo 2 OM - N.M - Motoso Mxn=-20 (Molos old km - Mosino o where. only the o component - 3 Bo sino - 24

un 12- x KM = Mo (sin & 7) + cos d'î) oux-2 plone , d = 0 only y component contribentes A. - Momoa? I sinollos o du Sind cos d'= - 1or Rel torticole Izur & & rel Ix-X11 l=0 u = l 22+1 Deti Y am 101, &') en d=1, m=1 will survive |x-x7 54 Comen Yu, C0V, ') /1001 2 Petting the values From @ 40 ineqe A = Mo () to rc arsine. - (-A) | - they mo o sina But Bo = Promo Mo = 2 o 30

As - Bool re sine 6 At t=0 At to the current in the coil is culoff. We have to prove that Ap = 3 Boa sime fertR? (A) A where v = I Но аг at to dt The deffusion equ i 2A = Moda 12A = 2 2 2 Art dr sind a cino a Ao) do do tl 22 A6 resin 20 dup A only depends on Ad Ap - Boo? 1 2 Sino (dronel

Now from G2A = I 22 Boor ve Sihod) I Boards & 82sing 2 r ade (- sin &* + los 87) DZA I Boa2 resino & function for re , snel ² A is The test Af er, 01 t) - Safino a CR-)J4 (KV) de = (stmo - simof ACK19 J, (6,1) de D'Arbogan (de acceso Ji (kir) sino ft Ir sine o sino e o jo are Alkit) J K 8)) Simof) tintimno bor (I di AIK) 5 (0) - Sino

from eq cih the third term of ② - - 1 and 2 sino a simo= 2 sindro - Cos (20) = 1-2 fine using & in 4 & ²A = Mod A A tri dit) = { "de A (8,0) e- J (Kr) ting | A$ = 3 Boa Sim of me -10% 5,5 ) de We have to prove that at to - 2ut R? Wm= 660 2 a3 [JCK) the force fel de xB & WM = Hor, 12 a lo, fue 2 + b²tulblod) Hu2+62-ulblos 3 UK = (5,4 b) po te pi Box 21

using from @ 4 w from We have Wm= 68. 2 a 3 p te [J, (K)] 2/6 - 2 utk