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Ampere’s Law Field zero?

Physics Asked on July 25, 2021

My textbook states that If $int Bcdot dl=0$, it does not necessarily mean that $B = 0$ everywhere along the path, only that the total current through an area bounded by the path is zero.

However later when calculating the magnetic field of a toroidal solenoid they say that the external field must be 0 because the enclosed current is 0. Isn’t that contradictory? The enclosed currents add to 0 but does that mean that the external field is 0? How is that possible if there is a stronger field contribution from the upper currents than the bottom currents so the net field shouldn’t be 0. Also for the interior of path 1 they say the field is 0 but it shouldn’t be 0 even if there is 0 charge enclosed because the magnetic field from the currents is still there right? In other words, it’s almost impossible to have a magnetic field of 0 when currents are present and the net current is 0. The only true time the field is 0 is when there are no currents at all is that right? So saying the field is 0 in paths 1 and 3 is false…

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Also another question I had is if instead it’s not currents in opposite directions but opposite charges (suppose negative on outside layer, positive on the inside layer) the electric flux is 0 outside (path 3) because the enclosed charge is 0. However, does that mean the field outside is 0? Isn’t the contribution from the outside layer stronger than the inner layer a distant r from the outer layer and so there is a net negative electric field. What about path 1? There is a field because of the positives charges and it radiates radially outwards. The net electric field is 0 but is it right to say that there are no electric fields?

Why do most textbooks seem to say that when the net enclosed charges/currents are 0 the fields must be 0? I understand for path 1 they might say the net electric field are 0, but for path 3 it doesn’t seem to make sense. So when someone says the electric fields from an electric dipole cancel, is that a false statement as there are still electric fields and a charged particle would feel a force?

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