Question

An electric field $\vec{E}$ is established across the ends of a cylindrical conductor of length L and area of cross-section A. Discuss how electrons attain an average velocity, independent of time. Hence, obtain a relation between current in the conductor and this ‘average velocity’ of electrons.

Hint:

Drift velocity $v_d$ is extremely small (around $10^{-4}\text{ m/s}$ or $0.1\text{ mm/s}$), whereas thermal velocity of electrons is very high (around $10^5\text{ m/s}$).

Answer 1

Concept:
• Free electrons inside conductor undergo frequent collisions with fixed heavy lattice ions.

• In absence of electric field, average thermal velocity is zero ($\vec{u}_{avg} = 0$).

• Electric field $\vec{E}$ exerts force $\vec{F} = -e \vec{E}$, producing acceleration $\vec{a} = \frac{-e \vec{E}}{m}$.

Step 1: Attainment of Time-Independent Average Drift Velocity
An applied electric field $\vec{E}$ accelerates free electrons opposite to field lines.
However, electrons do not accelerate indefinitely because they continuously collide with vibrating lattice ions.
In each collision, electron loses its directed momentum and resets its direction randomly.
The average time elapsed between two successive collisions is relaxation time $\tau$.
Velocity gained by $i$-th electron just before next collision:
\[ \vec{v}_i = \vec{u}_i + \vec{a} \tau_i \]
Averaging over all $N$ free electrons:
\[ \vec{v}_d = \frac{1}{N} \sum \vec{v}_i = \frac{1}{N} \sum \vec{u}_i + \vec{a} \left( \frac{1}{N} \sum \tau_i \right) \]
Since initial thermal velocities are completely random, $\frac{1}{N}\sum \vec{u}_i = 0$.
Defining average relaxation time $\tau = \frac{1}{N}\sum \tau_i$:
\[ \vec{v}_d = \vec{a} \tau = -\frac{e \vec{E}}{m} \tau \]
Magnitude of drift velocity:
\[ v_d = \frac{e E \tau}{m} \]
Since $e, E, m, \tau$ are constant at a given temperature, $v_d$ is a steady, constant average drift velocity independent of time.

Step 2: Relation Between Current and Drift Velocity
Consider cylindrical conductor of length $L$, cross-sectional area $A$, and free electron density $n$.
Total volume of conductor $V_{vol} = A \cdot L$.
Total number of free electrons in length $L$: $N_{total} = n A L$.
Total free charge in length $L$: $Q = (n A L) e$.
Time taken by electrons to drift through distance $L$: $t = \frac{L}{v_d}$.
Current $I$ flowing through conductor:
\[ I = \frac{Q}{t} = \frac{n A L e}{\frac{L}{v_d}} = n e A v_d \]

Step 3: Conclusion
Electrons attain steady drift velocity $v_d = \frac{e E \tau}{m}$ due to balance between acceleration and lattice collisions. The electric current is $I = n e A v_d$.