Here we will derive the equation showing the relationship between current and drift velocity. The drift velocity is the average velocity of the free charges and it is in the direction opposite to the electric field for electrons.
Drift velocity is quite small since there are so many free charges. If we have an estimate of the density of free electrons in a conductor, we can calculate the drift velocity for a given current. The larger the density, the lower the velocity required for a given current.
Derive the relationship between Current and drift velocity | Derivation of the drift velocity equation
For a conductor, the current I is given by:
I = nAvq
where A = area of cross-section of the conductor, n = number of charge carriers per cubic meter, q = charge on each charge carrier and v = drift velocity of charge carriers.
Note: for electron I = nAve, where e is the charge of an electron
This equation may be deduced as follows, with reference to Figure a:
● Consider a section of wire of length Δx
● The volume of wire in this section will be
ΔV = AΔx
If there are n charge carriers per unit volume in the wire, the number in volume ΔV will be
ΔN = nΔV = nAΔx
● If the charge on each charge carrier is q, the quantity of charge within the section will be
ΔQ = nAqΔx
● Suppose each charge carrier takes a time Δt to travel the distance Δx.
● Dividing both sides of the equation by Δt gives us
ΔQ/Δt = nAq Δx/Δt
● But ΔQ/Δt = I and Δx/Δt = v
● Hence I = nAvq
So this is the equation describing the relationship between current(I) and the drift velocity(v).
When the charge carrier is electron this equation can be rewritten as I = nAve, where e is the charge of an electron
As you have derived the current & drift velocity equation, now it’s time to solve few numerical problems using this formula. Here is the post you must visit to solve some selected problems.