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the diagram of scenarioIf I consider a finite length of wire in which current I is flowing. if I consider a circular loop above the finite wire, such that the current passing through that circular loop is zero. According to ampere's law. The current passing through that loop is zero and hence the magnetic field in that region of loop must be zero according to ampere law But by using biot savart law , the magnetic field in that region of loop is not zero

I am confused that by using ampere's law I get that the magnetic field is zero at that region but by using biot savart law I get magnetic field is not equal to zero.

I am explaining my situation through the diagram above

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    $\begingroup$ Maxwell's equations without current conservation are not internally consistent, so you can't have finite lengths of wire. You need complete circuits. $\endgroup$
    – mike stone
    Commented Jun 21 at 15:47
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    $\begingroup$ My answer here touches on why this happens. Briefly, the field derived from the Biot-Savart Law vanishes If and only if there are no "sources" or "sinks" of current. $\endgroup$
    – Michael Seifert
    Commented Jun 21 at 18:01

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The problem is that you have neglected an important part of your scenario, so that both of them are wrong, for different reasons. Specifically, with a finite current segment you will get charge accumulation at the ends of the segment. That means that the E field will be non-zero and will be changing with time. This changing E-field affects the results as follows:

A changing E field means that the Biot Savart equation is wrong. You cannot use it at all.

For Ampere's law the current through the loop is zero, but the "displacement current" is not zero. You cannot neglect the displacement current term, which is related to the changing E field.

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    $\begingroup$ This is how a dipole antenna works. As current flows from bottom to top the top becomes more positively charged and the bottom becomes more negatively charged. The longer the current flows the more charged the top and bottom become. The larger those charges become the stronger the E field. So the E field is changing in time $\endgroup$
    – Dale
    Commented Jun 21 at 18:30

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