# Why does a current carrying wire not have a higher current density on axis?

I am trying to resolve what is wrong with this line of thinking:

If I have DC flowing through a current carrying wire (a perfect cylindrical conductor) of radius $$r$$ - the magnetic field inside the wire is $$B_{\text{inside}}(r)= \mu_0Ir/2\pi R^2$$. Why does the Lorentz force produced by this magnetic field on the moving electric charges $$\vec{F}=\vec{v} \times \vec{B}$$ not push the electrons towards the axis - i.e. so that the current distribution is concentrated on axis?