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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?

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The external electric field isn't the only force acting on the electrons. They're also interacting with the fixed charges of the atomic nuclei in the conductor material.

The standard classical model for this is the Drude model, which assumes the interaction with the nuclei causes random scattering of the electrons. From this, Ohm's law can be derived.

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