* Turn the compass needle so it is approximately parallel to the wire. * Close the switch to send the current through the wire for about 5-10 seconds. * The compass will align itself with the magnetic field.

B Ampere’s circuital law right hand rule a I

current ==> magnetic field

a I B

a  B 0   < a a I B

a   a I B a  B

concentric hollow cylinders

0- 0 a b c  B

solenoid L +

we know that  B = 0 vector potential A we know that  [  x vector] = 0 we can now specify the vector let vector be A such that B =  x A William Thomson shows that Neumann's electromagnetic potential A is in fact the vector potential from which may be obtained via B =  x A.

vector potential A we also know  x B = µ o j B =  x A  x  x A  =  A) -    -      A = - µ o j is similar to Poisson’s equation but we have to solve three PDE’s A and j are in the same direction!!

j(r’) r’ A(r) r

R z’ r dz’ 2 L I A z

R z’ r dz’ 2 L I A z

Slide through the integral!

j(r’) r’ u r’ - r + B(r) r

R z’ r dz’ 2 L I B z

R z’ r dz’ 2 L I B z

summary jsjs Three techniques to find B 1] Ampere’s circuital law - lots of symmetry 2] find vector potential A, then B =  x A 3] Biot - Savart law B A BB