Question

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

Hint:

To easily memorize this fundamental derivation formula, remember the acronym name "NeAVd" (\(I = n e A v_d\)). It shows that current depends purely on the material properties (\(n\)), geometry (\(A\)), and electron kinetics (\(v_d\)).

Answer 1

Concept: Electric current (\(I\)) is defined macroscopically as the net rate of flow of electric charge passing through a given cross-sectional area of a conductor per unit time: \[ I = \frac{dq}{dt} \quad \text{or for uniform steady flow} \quad I = \frac{Q}{t} \] Drift velocity (\(v_d\)) is defined as the average uniform velocity with which free electrons get drifted inside a conductor opposite to the direction of an applied external electric field. We can establish the exact relationship by tracking the total charge contained within a geometric volume element of the conductor.

Step 1: Setting up the structural parameters of the conductor
Let us consider a uniform cylindrical metallic conductor characterized by the following physical properties:
• Let \(L\) be the total length of the cylindrical conductor.
• Let \(A\) be the uniform cross-sectional area of the conductor.
• Let \(n\) be the free electron density (also known as electron concentration or number of free electrons per unit volume of the conductor).
• Let \(e\) be the fundamental elementary charge of a single electron (\(e \approx 1.6 \times 10^{-19} \text{ C}\)).

Step 2: Calculating the total number of free electrons
The total geometric volume (\(V\)) of this uniform cylindrical conductor is given by the product of its cross-sectional area and its length: \[ \text{Volume } (V) = A \cdot L \] Since \(n\) represents the number of free electrons per unit volume, we can find the absolute total number of free electrons (\(N\)) contained within the entire volume of this conductor by multiplying the volume by the density: \[ N = \text{number density} \times \text{total volume} = n \cdot (A \cdot L) = nAL \]

Step 3: Calculating the total free charge available
Using the quantization of electric charge, the total magnitude of mobile electric charge (\(Q\)) carried by these \(N\) free electrons inside the conductor is given by: \[ Q = N \cdot e \] Substituting our expression for \(N\) from Step 2 into this equation yields: \[ Q = (nAL) \cdot e = nAeL \]

Step 4: Incorporating the drift velocity and time parameter
When an external voltage source is connected across the terminals of this conductor, an internal electric field is established. This field forces the free electrons to drift along the length of the conductor with an average drift velocity \(v_d\). The total time (\(t\)) required for all the free electrons initially tracking from one end of the conductor to completely cross out through the opposite terminal end over the distance \(L\) is given by the basic kinematic relation: \[ \text{Time } (t) = \frac{\text{Distance}}{\text{Speed}} = \frac{L}{v_d} \]

Step 5: Substituting charge and time into the definition of current
Now, substitute our explicit expressions for total charge \(Q\) (from Step 3) and total transit time \(t\) (from Step 4) into the fundamental definition of electric current: \[ I = \frac{Q}{t} \] \[ I = \frac{nAeL}{\left(\frac{L}{v_d}\right)} \] We can simplify this fraction by multiplying the numerator by the reciprocal of the denominator: \[ I = nAeL \times \frac{v_d}{L} \] Notice that the length parameter \(L\) appears linearly in both the numerator and the denominator. Therefore, it cancels out completely: \[ I = n e A v_d \] This completes the formal analytical derivation, establishing that the electric current is directly proportional to the free electron density, elementary charge, cross-sectional area, and the drift velocity.