3. * 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.

4. B Ampere’s circuital law right hand rule a I

5. current ==> magnetic field

7. a I B

8. a  B 0   < a a I B

9. a   a I B a  B

10. concentric hollow cylinders

11. 0- 0 a b c  B

12. solenoid L +

14. 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.

15. 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!!

16. j(r’) r’ A(r) r

17. R z’ r dz’ 2 L I A z

18. R z’ r dz’ 2 L I A z

20. Slide through the integral!

24. j(r’) r’ u r’ - r + B(r) r

25. R z’ r dz’ 2 L I B z

26. R z’ r dz’ 2 L I B z

28. 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