Biot Savart's law says there should be none as the current element is along the displacement vector, hence as $\sin \theta = 0$, so, the magnetic field must be zero.
But, as per Ampere's loop rule there is always some magnetic field even inside and on the conductor.
What is the resolution to this conflict?
[Physics] the magnetic field on surface of a current carrying conductor
electric-currentmagnetic fields
Best Answer
The r in the Biot-Savart integral has to be taken from every current element to P, and for all the current elements that don't lie on that top line in your diagram (since you mentioned in a comment that the current is supposed to be present throughout the 3-dimensional wire), r should not be parallel to dl, the direction of current for that element, so in that case the cross-product will be nonzero.
It might be interesting to consider the case of a 1D wire though--according to the equation here the magnetic field from an infinite straight 1D current should go to infinity as the distance from the wire approaches zero, but for a point along the wire, it seems the cross-product in the Biot-Savart integral should be zero for every current element, since r and dl are always parallel. Probably the resolution to the 1D problem is that the Biot-Savart integrand also has an $ |r|^3$ in the denominator, so for the current element that passes through the point where we want to evaluate the field, the integrand becomes undefined, so either you can't use the Biot-Savart law there or the field itself is undefined at that point in classical electromagnetism.