Doubt concerning biot savarts law Why do we calculate $dB$ for an infinitesimal part of a wire instead of a point?What is the reason behind that?Why can't we determine the magnetic field of a point with respect to a point of the wire instead of an infinitesimal part?
Also why is $dB$ proportional to $dl$ where $dl$ is the infinitesimal length of the wire?An infinitesimal quantity doesn't have any definite value,so how can we increase or decrease $dl$ for $dB$ to be proportional to it?Please give an intuitive explanation since it's really bothering me.
 A: A point cannot produce any magnetic field, it should have some length for it to produce any field.
Now coming to the second question. As pointed out by @Angry Refrigerator in their answer, dlXr shows the direction of the magnetic field, while I and distance from wire show its magnitude. It is not correct to say dB is proportional to dL. As you rightly thought, adding something to an infinitesimal number isnt correct and it is better for u to not think of it in that way. Maybe you can think of it qualitatively where length of wire doubled then mag. field double keeping everything same. But @Angry Refrigerator's interpretation is correct and think of it in that way.
A: As already mentioned by @Lili FN in the comments, "points" of a wire do not give rise to a magnetic field. But a small length of wire does.
This makes sense intuitively since it's actually the current (density) $\vec{j}(\vec{r})$ that gives rise to a magnetic field $\vec{B}$ (Ampere's Law).
And in the $1d$ case (infinitely thin wire), integrating over the direction of the current, or integrating over small bits of the wire $\mathrm{d}\vec{l}$ is equivalent.
Looking at the whole equation makes it easier to see why "$\mathrm{d}{B}$ is proportional to $\mathrm{d}{l}$". In the Biot-Savart law you're really integrating over the current at a given position $I(\vec{r})$. The $\frac{\mathrm{d}\vec{l} \times \vec{r}}{|r|^3}$ bit of the integral is just giving you a direction for the magnetic field $d\vec{B}$, while it's amplitude is dictated by $I(\vec{r})$. Just as Ampere's law hints at.
