On the singularity of Biot-Savart's law inside a current-carrying conductor When using Biot-Savart's law to compute magnetic flux density on a field point away from a current source point, the integrand is finite; however when using it to compute the field inside the source where "R" the distance between field and source is zero the integrand is singular.
All three versions, i.e. for linear currents, surface and volume currents this problem arises.
$$ B = \frac{\mu_0}{4\pi}\int_V \frac{J_v \times a_R}{R^2}dv $$
$$ B = \frac{\mu_0}{4\pi}\int_S \frac{J_s \times a_R}{R^2}ds $$
$$ B = \frac{\mu_0}{4\pi}\int_C \frac{Idl \times a_R}{R^2} $$
where $J_v$ and $J_s$ are the volume and surface current densities. $a_R$ is the unit vector pointing from source to field point.
Does the formula states that the magnetic flux density is infinitely large inside the source? Because this seems impossible.
I encountered this problem when I was trying to find the flux density on a surface current. The same issue happens when calculating the self-inductance using Neumann's formula.
 A: It's a typical problem in non-quantum electromagnetism.
This theory simply cannot describe self-interaction, so when you try to study the field too close to its source (in the point-like case in particular), the results may lose meaning. You're simply stepping outside the validity domain of the theory.
You can't always see it in the Biot-Savart integral, since an integral can be finite even if the function being integrated has singularities. But it's easy to find examples where the field, after computation, does have singularities near its source (wire, empty cylinder...)
However, be careful about another source of singularities that has nothing to do with self-interaction. In the context of classical electrodynamics, linear and surface currents are defined as the limit of a volumetric current when one or two of its dimensions are flattened to zero. This limit introduces artificial discontinuities.
For example, consider an infinite plane with a non-zero thickness $e$ with volumetric currents going in a single direction inside.

*

*If you compute the magnetic field it generates, you get a result with no singularity anywhere.

*Take the limit $e\to 0$ while maintaining the value of the current inside. You go from a volumetric to a surface model for the current, and the generated field become singular on the plane itself.

This singularity simply comes from the fact that you're neglecting the distance over which the field varies from one side of the plane to the other. It's called an artifact, because it doesn't mean anything physical, it's only a problem arising from your choice of model.
A: 
Does the formula states that the magnetic flux density is infinitely large inside the source? Because this seems impossible.

This situation occurs only when the current distribution is curve-like, with zero thickness (or point-like, due to moving charged point-like particle). Then magnetic field diverges at the line (or the point). This does not necessarily mean it is "infinitely large" at that line or point.
It is "infinitely large" in the sense its magnitude diverges to infinity when one approaches to the current line or point from any direction. But there is no single limit, because direction of magnetic field always depends on the direction of approach. This is a mathematical fact. It is not correct to say this is impossible in mathematics.
This means that we have a discontinuous magnetic field, not infinite magnetic field. In fact, magnetic field definition can be sometimes completed even in that point of singularity. For example, if there is other source of magnetic field, this acts on the line/point, and we can define total magnetic field at the line/point as the external magnetic field due to the other source.
If the current distribution is planar, i.e. current is running in a plane, then there may not be a divergence at all. If finite line element in the plane, perpendicular to the current, is associated with finite current passing through it, then even infinite plane with net infinite current will create finite magnetic field in both half-spaces. This follows from Ampere's law; magnetic intensity times length of a line segment equals net current intensity associated with that length.
