Question

A cylindrical conductor of length l and cross-section area A is connected to a DC source. Under the influence of an electric field set up due to the source, the free electrons begin to drift in the opposite direction of the electric field.

1. Draw the curve showing the dependency of drift velocity on relaxation time.
2. If the DC source is replaced by a source whose current changes its magnitude with time such that I = I₀ sin 2πvt, where v is the frequency of variation of current, then determine the average drift velocity of the free electrons over one complete cycle.

Answer 1

Part 1: Drift Velocity vs Relaxation Time Curve

The drift velocity vdv_dvd​ of free electrons in a conductor is given by the formula: `v_d = (eEr)/m`

Graph:

Part 2: Time-Varying Current I = I₀ sin⁡(2πvt)

The drift velocity is also related to current I by:

`v_d = I/(nAe)`

`I(t) = I_0 sin(2pivt) => v_d(t) = I_0/(nAe) sin (2pivt)`

Average Drift Velocity Over One Cycle:

`barv = 1/T int_o^T v_d(t) dt = 1/T xx I_0/(nAe) int_0^T sin(2pivt) dt`

`int_0^T sin (2pivt) dt = 0`

The average drift velocity over one complete cycle is 0