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vd = a τ
where α is the acceleration due to the electric field
τ is the average relaxation time.
Now
Acceleration of each electron is, α = -eE/m
=> vd = -eEτ/m
Electric field in a conductor doesn't change for a given value of potential V.
At constant temperature, average relaxation time remains unchanged.
Therefore, the drift velocity remains constant for constant temperature.
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