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Suppose you have a circle and for any $da$ in the circle you have a wire going through it with current $I\cdot da$. If you take an ampere loop in the shape of a circle centerd around the center of the circle and with radius r you get the equation: $2\pi rB=\mu_{0}I\pi r^{2}$

which implies $\boldsymbol{B}=\frac{\mu_{0}Ir}{2\pi}\hat{\theta}$

however you can get the same result with any circle regardless of it's radius and center but the magnetic field cant be diffrent, so where is my mistake?

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  • $\begingroup$ It is $j.da$ ($j$ density of current) instead of $I.da$ isn't it? $\endgroup$ Mar 23, 2021 at 19:47
  • $\begingroup$ I dont know but i just needed it to be the case that any integral of the current is proportional to the Area of the domain $\endgroup$ Mar 23, 2021 at 20:02

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you can get the same result with any circle regardless of it's radius and center but the magnetic field cant be diffrent, so where is my mistake?

To find the B-field around the circle with the current through the center of the circle you knew, because of the symmetry of the physical arrangement, that the B-field (in the direction tangential to the circle) would be equal at all points on your circle.

When you move the circle so the current no longer passes through its center you no longer have equal B field at all points on the circle. You will still get the same path integral of B-field around the circle, but you will need some other logic to tell you what it is at each individual point on the circle.

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