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Ampere’s circuital law vector potential Biot - Savart

Ampere’s circuital law vector potential Biot - Savart. b. * 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. a. I.

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Ampere’s circuital law vector potential Biot - Savart

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  1. Ampere’s circuital law vector potential Biot - Savart b

  2. * 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.

  3. B a I Ampere’s circuital lawright hand rule

  4. current ==> magnetic field Stokes

  5. positive charge out of screen B

  6. B a I

  7. B a r B a I 0  r < a

  8. B a r B a I a  r

  9. concentric hollow cylinders

  10. B 0- 0 a b c r

  11. • + L solenoid superposition

  12. Thomson Lord Kelvin 1824-1907

  13. 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.

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

  15. j(r’) A(r) r r’

  16. z dz’ 2 L z’ R A r I

  17. z dz’ 2 L z’ R A r I after the integration

  18. Biot 1774-1862 Savart 1791-1841 Biot-Savart law

  19. Slide through the integral! Biot-Savart law

  20. Biot-Savart law

  21. 0 Biot-Savart law

  22. Biot-Savart law

  23. j(r’) B(r) + r’ r ur’ - r Biot-Savart law

  24. z dz’ 2 L z’ R B r I

  25. z dz’ 2 L z’ R B r I

  26. magnetic dipole

  27. B B B js A summary 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

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