Minor Question about Biot-Savart Law Formulation

Question

On Wikipedia, it is stated that the Biot-Savart law is a formula used for determining the magnetic field pseudovector B(r) at a position vector r due to a current-carrying wire, but that the formula only works for constant/steady electric currents. Wikipedia then states the law:

So my question then is, if the magnetostatic assumption is employed, would it be okay to factor that current term $I$ out of the line integral? And if it is legitimate, why do most textbooks leave it in the integrand? Is there some hidden benefit to doing so?

Answer 1

If the electric current is confined to a 1D wire with current $I({\bf r})$, then its divergence $\nabla \cdot {\bf J} = dI/dl$, where $l$ parameterizes the arc length along the wire. By the continuity equation, this equals $-\partial \rho/\partial t$. So under the electromagnetostatic assumption, $I$ must be constant along the wire and we can indeed pull it outside of the integral. But in principle, you could have a magnetostatic but electrodynamic situation in which the $I$ needs to stay inside of the integral.

But even though the current is generally constant along the wire in practice, it's conventionally left inside of the integral because that makes it easier so see why the product $I\, dl$ generalizes to $J\, dV$ in the case of bulk currents. For bulk currents, the $J$ generically varies over space even in the electromagnetostatic case, so you need to leave it inside of the integral.

Answer 2

It is not fine to factor the $I$ out of this integration. While the magnetostatic assumption implies that $I$ will not depend on time, that's no reason to suspect that the current does not depend upon the location along the curve you're integrating along. That is, while the current is assumed to not vary in time, it may still vary in space.