I have the formula:
$\vec{J}=\sigma \vec{E}$
Here, $\vec{J}$ is the current density through a wire, $\sigma$ is the conductivity of the material, constant throughout, and $\vec{E}$ is the electric field that moves current along.
The above equation implies that if $\vec{E}$ is constant for the unit cross-section, and all throughout the wire, then $\vec{J}$ will be constant all throughout the wire.
This seems to contradict Newton's laws. If you have a constant force acting on a particle, it is a given that it will accelerate. If an electric field $\vec{E}$ acts on a set amount of charge that goes through the cross-section per unit time, further down the wire it should be going faster and hence the current density $\vec{J}$ would be greater.
What is the explanation?