Question

Define Drift Velocity and establish its relation with electric current.

Hint:

Important formulas: - Drift velocity: \( v_d = \mu E \) - Current relation: \( I = nqAv_d \) - Even though drift velocity is small, current appears instantly due to electric field propagation.

Answer 1

Concept: In a conductor, free electrons move randomly due to thermal energy. When an electric field is applied, these electrons acquire a small net velocity in a particular direction. This average velocity is called
drift velocity.
Definition: Drift velocity is the average velocity acquired by charge carriers in a conductor under the influence of an external electric field. It is denoted by \( v_d \).
Step 1: Motion of Charge Carriers Consider:

• A conductor of cross-sectional area \( A \)
• Number of charge carriers per unit volume = \( n \)
• Charge of each carrier = \( q \)
• Drift velocity = \( v_d \)

In time \( t \), charge carriers move a distance: \[ \ell = v_d t \]
Step 2: Volume of Charge Flowing Volume of conductor through which charges move: \[ \text{Volume} = A \times \ell = A v_d t \] Number of charge carriers in this volume: \[ N = n A v_d t \]
Step 3: Total Charge Flow Total charge crossing the section in time \( t \): \[ Q = nqAv_d t \]
Step 4: Relation with Electric Current By definition of current: \[ I = \frac{Q}{t} \] Substituting \( Q \): \[ I = \frac{nqAv_d t}{t} \] \[ \boxed{I = nqAv_d} \] This is the required relation between current and drift velocity.
Special Case: Electrons as Charge Carriers If electrons are charge carriers: \[ q = e \] \[ I = neAv_d \] Direction:

• Electron drift is opposite to electric field
• Current direction is opposite to electron motion

Drift Velocity in terms of Electric Field From microscopic theory: \[ v_d = \mu E \] where \( \mu \) is mobility. Thus: \[ I = nqA\mu E \] This leads to Ohm’s law at microscopic level.
Key Observations:

• Current depends on number density of charge carriers.
• Larger cross-section gives larger current.
• Drift velocity is very small compared to speed of light.