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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 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 Ampere’s circuital lawright hand rule
current ==> magnetic field Stokes
positive charge out of screen B
B a I
B a r B a I 0 r < a
B a r B a I a r
B 0- 0 a b c r
• + L solenoid superposition
Thomson Lord Kelvin 1824-1907
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.
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!!
j(r’) A(r) r r’
z dz’ 2 L z’ R A r I
z dz’ 2 L z’ R A r I after the integration
Biot 1774-1862 Savart 1791-1841 Biot-Savart law
Slide through the integral! Biot-Savart law
0 Biot-Savart law
j(r’) B(r) + r’ r ur’ - r Biot-Savart law
z dz’ 2 L z’ R B r I
z dz’ 2 L z’ R B r I
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