Question

9. (20 points) Suppose you have an infinite sheet of thickness d, the bottom lying at y = 0 in the x-z plane, that carries a current density described by ] = J. ł, as shown in the image below. d j = Jože ---- I. (4 points) Describe this current density (i.e., it's direction in the image and how it is distributed in the slab). (16 points) Find the magnetic field, B, everywhere. II.

Answer 1

Given, e consider the co-ordinate system as shown in figure o anfinate sheet of thikness a leging at 4=0 in the nay plane. Current density is given as T = Not să & Ken Coctab) - The driction of current density is positive driction of s-aus - current density is varying lenairly along 7-anis Upto 7=bi - planes parrell to (-y) plane have some current density across them. J = 0 (Z4 AT) e-yplane -T, and I are constant across their respective planes.

Eldydy 9 -digde feild at - Magnatic point p area let us take a small of cross section dydz Bileno Biltino it indicates a long infinale wire parrell to x-anis with cross section area a. and fild due to it is Bi. let us contida an anti-Element for above one, setch that 18,1 = 17,J an to 1B,1-1€) A but the driction of Bi and By are different. - By the symmetry vertical Componants of B and By are Cancel Each other as showe in figure - 1B | Loso Componats along y-anis semain. i. The feild at any point above the infinate theet is the y-amis, and similarly to below the infinate sheet is - vey-anis.

TEB Case (i) (x > blant (720) al = 1 Bu Fut Amphers Loop e lalong the path and is same due to cylindrical Symmetry WBT from Ampher low, Bide = el calored Enchon I Tidal - Bidet Bide + B. d + (Biden from figure Blade Perched - Solat - Sold - B des de Bodi = B. Il=0 and wish Bil= 1 Bul=B Bode = Blloso = Be BA+BA = llo sosh Bridly Beloso = Be B = loud E -slojd led) old; (izzo)

case & Colzed) ØB de lo lanedland tenilen - Soda = Sig a dy-tz # d- El sot dy-dz Od I amphes! I am loop - from above part e.de = 2B4 24 200 X (z?- ) B = Mosol (d - z) = 72) 1 B = dosol (d-z) » _ 22) COLELd/ 40 By obsencing term ((d-z) -42 so if 72412 so y Esdl, B20 for Ext/2 o for 72d/ BIO