Why Does Constant DC Current Not Imply Changing Electron Velocity?

[yabb dabba do]

dV/dx=E
E*q=F=m*a
dq/dt=I
v/R=I

Say you have a situation where you have a constant DC current in a conductor. Applying the above equations to this situations seems to lead to a contradiction, so I’m confused.

-So if you have constant DC current then that indicates dq/dt is a constant
-If dq/dt is a constant the velocity of the electrons is not changing
-However in order to have a current you need a voltage difference and if you have a voltage difference you have an electric field.
-If you apply an electric field applied to electrons, you have a non-zero force on them. The force indicates that the electrons are accelerating.
-If the electors are accelerating this indicates their velocity is not constant, and hence dq/dt would not be a constant, which contradicts the first statement above.

Where did I go wrong in my reasoning?

 

[anorlunda]

Look up the Drude Model of conduction on Wikipedia.

 

[Baluncore]

Say you have a situation where you have a constant DC current in a conductor.

Conductors have resistance, that is the reciprocal of the conductance. An electric field is needed to move current through the resistance. The voltage drop along the conductor, multiplied by the current, is the power generated as heat.

 

[tech99]

If you imagine the electron as a car driving along a road, it required some force (axial E-field) to accelerate it first of all, but in a perfect vacuum, in a frictionless world, it would require no force (axial E-field) to keep it going. This is what Newton said. In practice, the car experiences some air resistance and friction, equivalent to electrical resistance, so there is a need for a continuing small force (axial E-field) to keep it moving at constant velocity.
During the initial acceleration, by the way, the inertia of the electron is not only created by its mass, but also its need to build a magnetic field as it gains velocity. When we try to stop the electron, the energy stored in its KE and in its magnetic field is given back to us, and is seen in the form of a forward voltage kick, which will often do work in the form of a spark.