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Consider a circular loop of radius r and a wire of finite length which lies along the axis of the loop. Current I flows through the wire and.
I am trying to to find $\int \vec B \bullet d\vec l$ over the circular loop.
If I find magnetic field at a point on the loop And integrate it I get a non-zero answer but when I use Ampere's Law
$\int \vec B \bullet d\vec l =\mu I$ here the current which pierces the loop is zero so the integral turns out to be zero.
Why does Ampere's law give a different answer?

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  • $\begingroup$ You are simply making the mistake that $\int B\cdot d\ell = 0 \rightarrow B=0$, which is not necessarily true $\endgroup$
    – Triatticus
    Jul 12, 2020 at 17:45
  • $\begingroup$ How do you account for the current in the wire? I think if you consider charge accumulation at the ends of the wire you would end up with the desired result. $\endgroup$
    – srswat
    Jul 12, 2020 at 18:37

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The Ampere's law is valid only for steady currents and so when you use the Ampere's law in between the loop and piece of wire, you won't get the correct result because the current in the piece of wire isn't steady.

You see charge keeps accumulating on one side of the wire and hence the current isn't steady.

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We can't apply ampere circuital law in case of finite length wire . Ampere circuital law is applied in case of a wire with infinite legth because infinite wire is considered to be a closed circuit . Finite wire is not a closed circuit and the is "Ampere Circuital law" so it can be applied in a closed circuit only .

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