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Drift Velocity

Drift Velocity :- In the absence of applied potential difference, electrons have random motion and have only thermal velocity. When a battery is connected across the terminals of the conductor, then in addition to thermal velocity, they acquire an acceleration due to applied electric field. Now the electrons acquires a velocity component in a direction opposite to the direction of the electric field. So

“Drift velocity is defined as the average velocity with which the free electrons move towards the positive terminal of the battery under the effect of the applied external electric field.”

The gain in velocity due to the applied field is very small and is lost in the next collision. Due to this the battery has to constantly supply energy (kinetic energy) to the electrons for the flow of constant electric current.

Expression for drift velocity

Let the two ends of a conductors are joined to a battery. Then one end is at higher potential and another at lower potential. This produces an electric field E inside the conductor from point of higher to lower potential.

Force experienced by each electron, $\displaystyle \overrightarrow{F}=-e\overrightarrow{E}$

Acceleration of electron, $\displaystyle \overrightarrow{a}=\frac{\overrightarrow{F}}{m}=\frac{-e\overrightarrow{E}}{m}$

At any time, velocity of an electron is given by, $\displaystyle {{\vec{v}}_{1}}={{\vec{u}}_{1}}+\vec{a}{{\tau }_{1}}$

where u₁ is the thermal velocity, τ₁ is the relaxation time(the time that has elapsed since the last collision) and aτ₁ is the velocity acquired by the electron under the influence of the applied electric field.

Similarly, the velocities of the other electrons are :-

$\displaystyle {{\vec{v}}_{2}}={{\vec{u}}_{2}}+\vec{a}{{\tau }_{2}}$

$\displaystyle {{\vec{v}}_{3}}={{\vec{u}}_{3}}+\vec{a}{{\tau }_{3}}$

. . . . . . . . . . . . . . .

$\displaystyle {{\vec{v}}_{N}}={{\vec{u}}_{N}}+\vec{a}{{\tau }_{N}}$

The average velocity of all the free electrons in the conductor :-

$\displaystyle {{\vec{v}}_{d}}=\frac{{{{\vec{v}}}_{1}}+{{{\vec{v}}}_{2}}+{{{\vec{v}}}_{3}}+...{{{\vec{v}}}_{N}}}{N}$

$\displaystyle {{\vec{v}}_{d}}=\frac{({{u}_{1}}+\vec{a}{{\tau }_{1}})+({{{\vec{u}}}_{2}}+\vec{a}{{\tau }_{2}})+...+({{{\vec{u}}}_{N}}+\vec{a}{{\tau }_{N}})}{N}$

$\displaystyle {{{\vec{v}}}_{d}}=\frac{({{{\vec{u}}}_{1}}+{{{\vec{u}}}_{2}}+...+{{{\vec{u}}}_{N}})}{N}+\vec{a}\left( \frac{{{\tau }_{1}}+{{\tau }_{2}}+...+{{\tau }_{N}}}{N} \right)$