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## Ampere’s circuital law vector potential Biot - Savart

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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