Explanation of Biot-Savart Law

[JustinoChino]

I've learned that in a wire with a current flowing through it, a magnetic field is produced, and that to determine the direction of the fields, one could match their thumb with the direction of current and curl their fingers around the wire as shown in the link below. I also learned that in Biot Savart's Law, the magnetic field at a given point P is equal to the summation of infinitesimally small magnetic fields resulting from current flowing through infinitesimally small wire segments. How can this be possible in a straight wire if the magnetic field lines are always perpendicular to the current flow? If an infinitesimally small segment of a wire is at an angle to the point P, how is it contributing to the net field at that point?

https://upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Manoderecha.svg/220px-Manoderecha.svg.png

 

[BvU]

Hello Justino,

If an infinitesimally small segment of a wire is at an angle to the point P, how is it contributing to the net field at that point

According to $\ \displaystyle {I\;d{\bf \vec L}\times \hat{\bf r}\over r^2}\ \$, a vector product that ensures that the field is perpendicular, that the angle matters and that further away current contributes less