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. a I
r  a
a r B

6. a I
r  a
a r B

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

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

11. R z’ r
dz’
2 L I A z

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

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

14. R z’ r
dz’
2 L I B z

15. magnetic dipole

16. large hadron collider

17. Earth’s magnetic field protects us

18. Inductance L

19. a b
Coaxial cable

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

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 ( ).

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

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

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

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

28. Faraday’s law
apply Stoke’s theorem

30. wire carrying current IL u DV 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 c u

33. B X c u
V12= 0

34. B X u c
V12= Bcu

35. B X u c
V12= Bcu

36. B X c u
V12= Bcu

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

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

39. I1 dl1 1 B1

40. I2 dl2 1 B2

41. I1 dl1
I2 dl2 1 B2 B1

42. I1 dl1
I2 dl2 1 B2 B1

43. I induced

44. electromagnetic
launcher

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

46. before

47. before
after