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Why don't electrons accelerate when a voltage is applied between two points in in a circuit? All the textbooks I've referred conveyed the meaning that when an electron traveled from negative potential to positive potential, the velocity of the electron is a constant.

Please explain.

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You are asking why is there no force on an electron from a voltage difference, i.e. from the electrostatic potential between the two terminals of your circuit. Any resulting force is proportional to the gradient of the potential. –  Chris Gerig Dec 27 '11 at 4:22
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5 Answers

You may consider the relation: $$i=nAev_d$$ $$=>v_d=\frac{i}{nAe}$$ $A$ : Cross -Sectional Area

$n$ : Concentarion of electrons

$v_d$ : Drift velocity

$i$ : Current

If a constant current flows through a conductor of varying cross section the drift velocity will change

In fact we have the relation:$$j=\sigma E$$ If the cross section changes[current remaining constant ] the current density,$j$ will change. Consequently E will change if the conductivity $\sigma$ is constant[for a homogeneous material with constant values for n and $\sigma$].

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At the classical level the explanation provided in the previous answers is known as the Drude model. There is additional info on Wikipedia: http://en.wikipedia.org/wiki/Drude_model

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And one can add that they are accelerated if the circuit element and the voltage difference is the one applied on a vacuum tube, the simplest particle accelerator. In the vacuum there is no resistance and statistical transfer of energy to other electrons.

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Yes, the electron is accelerated by the external electric field $E$, but at the same time it is "decelerated" with collisions with obstacles. These collisions are modelled as a "friction" force proportional to the electron velocity, something like this: $$m_e\frac{dv}{dt} = eE-r\cdot v$$ This equation has a quasi-stationary solution when the dragging force cannot exceed the resistance force: $$eE=r\cdot v$$ This gives a constant (average or drift) velocity. This picture is literally applicable to the gas discharge (current in a gas) where the electrons are particles accelerated between collisions with atoms.

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Electrons are accelerated by the constant applied electric field that comes from the external potential difference between two points, but are decelerated by the intense internal electric fields from the material atoms that makes up the circuit. This effect is modeled as resistance.

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