Question

Establish the relation between drift velocity of electrons ($v_d$) and electric current ($I$) in a conductor.

Hint:

You can remember this formula using the mnemonic "I = venA" (current equals velocity, charge, number density, Area) or just "$neAv_d$". This relation is crucial for deriving Ohm's law from microscopic principles.

Answer 1

Concept:
• When an electric field is applied across a conductor, the free electrons drift towards the positive terminal with an average velocity known as the drift velocity ($v_d$).
• The macroscopic electric current ($I$) flowing through the conductor is fundamentally derived from this microscopic drifting motion of the electrons.

Step 1: Define the parameters of the conductor
Consider a cylindrical conductor of length $L$ and uniform cross-sectional area $A$.
Let $n$ be the number density of free electrons, which is the number of free electrons per unit volume.
The total volume of the considered conductor segment is $V_{vol} = A \times L$.
Therefore, the total number of free electrons in this segment is $N = n \times A \times L$.

Step 2: Calculate the total mobile charge
Let $e$ be the magnitude of the charge of a single electron.
The total charge $Q$ contained within this section of the conductor is:
\[ Q = N \times e \]
\[ Q = (n A L) e = n e A L \]

Step 3: Relate charge to current using drift velocity
When a potential difference is applied, all these free electrons drift with an average velocity $v_d$.
The time $t$ required for an electron to travel the entire length $L$ of the conductor is given by:
\[ t = \frac{L}{v_d} \]
Electric current $I$ is defined as the rate of flow of electric charge across a cross-section:
\[ I = \frac{Q}{t} \]
Substitute the expressions for $Q$ and $t$ into the current equation:
\[ I = \frac{n e A L}{\left(\frac{L}{v_d}\right)} \]

Step 4: Simplify the final expression
The length parameter $L$ cancels out from the numerator and denominator:
\[ I = n e A v_d \]

Step 5: Conclusion
The relationship between electric current and drift velocity is established as $I = n e A v_d$. This shows that current is directly proportional to the drift velocity.