derive an expression for drift velocity of an electron in a conductor

The expression for the drift velocity vdv_dvd​ of an electron in a conductor can be derived as follows:

1. When an electric field EEE is applied across a conductor, the force FFF on an electron (charge −e-e−e) is:

F=−eEF=-eEF=−eE

2. Using Newton's second law, the acceleration aaa of the electron is:

a=Fm=−eEma=\frac{F}{m}=\frac{-eE}{m}a=mF​=m−eE​

where mmm is the mass of the electron.

3. Electrons in a conductor experience collisions at intervals, characterized by the relaxation time τ\tau τ, which is the average time between collisions.
4. The drift velocity vdv_dvd​ is the average velocity gained by an electron under acceleration during the time τ\tau τ:

vd=aτ=−eEmτv_d=a\tau =\frac{-eE}{m}\tau vd​=aτ=m−eE​τ

This means the electrons drift opposite to the direction of the electric field. In summary, the drift velocity is given by:

vd=−eEτm\boxed{v_d=-\frac{eE\tau}{m}}vd​=−meEτ​​

Where:

• eee is the magnitude of the electron charge,
• EEE is the applied electric field,
• τ\tau τ is the relaxation time,
• mmm is the electron mass.

This is the fundamental expression for the drift velocity of electrons in a conductor under an applied electric field.