Question

Define the term 'drift velocity' of electrons in a current carrying conductor and derive, in its term, the expression for current flowing through the conductor.

Hint:

Drift velocity is the average velocity of free electrons opposite to the field. Count the charge crossing area A in time t to get I = n e A v_d.

Answer 1

Step 1: Define drift velocity:
Inside a conductor, free electrons move randomly. When an electric field E is applied, they pick up a small average velocity opposite to the field.
This average velocity of the free electrons in the direction opposite to the applied field is called the drift velocity \( v_d \). Its value is \( v_d = \dfrac{eE\tau}{m} \), where \( \tau \) is the relaxation time.

Step 2: Set up the conductor:
Take a conductor of cross-section area A. Let n be the number of free electrons per unit volume. Each electron has charge e.

Step 3: Charge crossing a section:
In time \( \Delta t \), all electrons within a distance \( v_d \Delta t \) behind the section cross it.
The volume of this part is \( A v_d \Delta t \), so the number of electrons is \( n A v_d \Delta t \).
Charge crossing: \( \Delta Q = n A v_d \Delta t \cdot e \).

Step 4: Current:
\[ I = \frac{\Delta Q}{\Delta t} = n e A v_d \]
The current is directly proportional to the drift velocity.

Final Answer:
\( I = n e A v_d \). \[ \boxed{I = n e A v_d} \]