Question

2. Consider a static volume current density J(r') where r' is the position vector of a point in the current distribution. Show that the field generated at a point with position vector r, according to the Biot-Savart law, u mrJ(r')RJ B(r) = -JJJp3 av, 477 o in which R=r-r' and R=R , satisfies Maxwell's magnetostatic equation V x B = 4J (u should be considered as constant). Consider the magnetic vector potential defined by A and the Lorenz gauge Show that this potential satisfies the equation B = V condition V. A=0. [5] (Hint: see Useful formulae and apply the equation of continuity and anti-symmetry under interchange of r,r', where appropriate.]

Answer 1

ANSWER

Given that
A statłe volume current densì ky J(y) Y = position vector B a point in the current Hen e dis bribulon. accord ing the Bìob -Savar t law- R3 I1) XR 417 R8 Where R= 91-9 and R= IRI 19= positlon vector. "Th is Eruation should follow Naxwell's nagnetostatiC Erug tion given as VXB=UJ. The ELuabion O can be AcuYiten a BLY)= 4 See that the biob-savart's law is an improper fntegl we we need to solve this by saying the integr al is the cauchy's principal valuc which ntegral is the defined in Lerms of. a limit (we make a substitutIOn), 19-713

BLY)= - u using standaad vector Calculus iekl- Pdentity we get Bl)= 471 Taking the curl opeator outside the întegral smce both tmits the curl and integral are defIned in lerms & T(al) d3V B(Y) = Vn X 19-31) Applying url to both stdes g the gruation and Ehe vector identitg for the curl of a curl we get VaX Bl0) 47 VJ ADw since the 1St Lerm is O as we know th et VJ(a1) =-JRN lato (con tintata Gauato) Dg XBlo1)= TO Evaluate thìs, we apply diver gence theorem.

శ I=-417 Putting this E3uation to Esu B we get This shows that it satisfies Naxwell's Nagnetostatic Esuation. Now for ther we have the Magnetic vector pobenttal gìven as, A- v- we need to show that B=VXA and v. A =0 () proof for B= VXA Condition !- we can sietort te the Nagnetic vector Eeuation as, A= la-a1 Taking cual ot this we get J(a1) d3y V, х

Vn XA dy . This Eruatton is 82ua) to the suatlon & B a8 ean been seen tn Ezuh VXA =B (11) PYoof for 2onenz "Gauge condition v-A =0 "Taking divergenee & Esu'n @ we geb. ひA= A V- 419 1a-a1). Actording to continuitg Eruation D.J = - aRylat. afvlat =0- But Smce fîn this case Hence v-A =0,