If we consider a simple dc circuit with just a resistor, we notice that V is constant and so is the length through which current is flowing, hence the electric field is constant. So why does drift velocity also remain constant shouldn't it be increasing linearly?
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Why do you think drift velocity should be increasingly linearly ? In a resistive material the drift velocity $u$ is proportional to the electric field $E$, so if $E$ is constant so is $u$ (ignoring any transient state when the current is first turned on). See the Wikipedia article on drift velocity for further details.
gandalf61
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case 1. no electric field: net drift velocity is 0. case 2. there is an external and constant electric field and electrons move with net constant acceleration. Can't we consider the question to be a superposition of the above two cases?(since there is no motion of positive charges) and so wouldn't the electrons be accelarating? – Apr 07 '20 at 11:12
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@Jonathan You are picturing the electrons as if they were moving in a vacuum, which has zero resistivity. But in fact the electrons are moving in a wire, which will have a non-zero resistivity. The resistivity of the wire (which is due to a combination of impurities in the metallic lattice and thermal vibrations of its atoms) opposes the electrons' motion and the end results is that the electrons have a maximum drift velocity which has a finite value. – gandalf61 Apr 07 '20 at 11:28
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So our understanding that ratio of drift velocity to electric field is constant is experimental? – Apr 07 '20 at 11:57
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@Jonathan Yes. It is a consequence of Ohm's Law, which is an empirical law. – gandalf61 Apr 07 '20 at 12:11