According to london's first equation,electrons continously accelerate,when electric field is applied across a conductor,this way electrons can acheive speed greater than that of light,but a particle can't acheive speed greater than light.
In free space also the electron will accelerate on application of electric field that does not mean that electron will gain velocity greater than light. This is what is done in a synchrotron, for example. As the velocity of speed of electron reaches close to that of the speed of light, the same amount of force does not result in the same amount of acceleration. The end result is that electron never reaches the speed of light.
The electron will reach the speed close to speed of light (around 99%) but does not break the speed barrier. If we further apply the force, the energy converts to mass, i.e. mass of the electron increases. We cannot cross the light barrier.
The acceleration of electrons will tend to zero with the passage of time and electrons will start moving with a constant velocity, its relation with time Or the acceleration will decrease and will still remain positive.
p = ymv (read y as gamma).
The more force you apply, the larger the momentum will be, however, the velocity will not increase at the same rate (i.e. with the same acceleration).
Why the velocity will not increase with same rate?
F = dp/dt -------- (1)
p = ymv -------- (2)
you should get F = (y^3)m*dv/dt.
Since F is constant, increase in v (with time v will increase due to acceleration) leads to increase in y and hence decrease in the value of dv/dt.
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