static magnetic fields

1. static magnetic fields

2. Static magnetic fields • Charge in motion yields a current I • I = j  area • j is a vector -- current density -- amperes/meter2 • Ampere’s circuital law  B  dl = o Ienc

3. 1 Tesla = 104 Gauss • B at equator  1 Gauss

4. current ==> magnetic field

5. B a a r I r a

6. B a a r I r a

7.  B  dl = o Ienc • due to one wire • Ienc = I •  B  dl = B [2 p r] • B = o {1 / 2 p r} I • due to other wire • B = o {1 / 2 p r} I • superposition

8. L • Ienc = N I •  B  dl = B [2 L ] • B = o N I / 2L • B = o N I / L -- in center / top and bottom

9. vector potential A we know that • B = 0 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.

10. vector potential A B =  x A we also know  x B = µoj  x  x A  - A = • A) - 2A  A = - µoj is similar to Poisson’s equation but we have to solve three PDE’s A and j are in the same direction!!

11. z dz’ 2 L z’ R A r I

12. z dz’ 2 L z’ R B r I after the integration

13. z dz’ 2 L z’ R B r I Biot-Savart integral

14. z dz’ 2 L z’ R B r I

15. magnetic dipole

16. large hadron collider

17. Earth’s magnetic field protects us

18. Inductance L

19. b a Coaxial cable

20. d I z w B I Inductance of a microstrip

21. time-varying magnetic fields

22. An induced electric current flows in a direction such that the current opposes the change that induced it. This law was deduced in 1834 by the Russian physicist Heinrich Friedrich Emil Lenz (1804-65).

23. Faraday’s law either B or s individually change in time or they both change in time together

25. z B ds y x a a magnetic field changes in time

26. z B ds y x size of loop changes in time

27. z B y x L R w j size of loop changes in time

28. Faraday’s law apply Stoke’s theorem

30. 2w L u I DV wire carrying current I nonuniform B field

31. Bewley’s book • trick questions • not every motion generates a voltage • uniform B & v • substitution of circuit • Vgen = 0!

32. X B u c 1 2

33. u c 1 2 X B V12= 0

34. X B u c 1 2 V12= Bcu

35. X B u c 1 2 V12= Bcu

36. u c 1 2 X B V12= Bcu

37. X B u c 1 2 V12= Bcu V12= -Bcu

38. X B u c 1 2 V12= Bcu V12= -Bcu

39. I1 dl1 B1 1

40. I2 dl2 B2 1

41. I1 dl1 I2 dl2 B2 B1 1

42. 1 I1 dl1 I2 dl2 B2 B1

43. I induced

44. electromagnetic launcher

45. J. Phys. D 33, 120 (2000)

46. before

47. before after