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Question
Derive an expression for resistivity of a conductor in terms of the number density of charge carriers in the conductor and relaxation time.
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Solution
The relationship between the relaxation time (τ) and drift velocity `(v_d)` is given by:
`v_d = -e((Eτ)/m)`
∴ `τ = ((v_d m))/e xx E`
Let L = Length of the conductor
A = Area of the conductor
n = free electron density
e = charge of the electron
E = Electric field
m = mass of the electron
τ = Relaxation time
The current flowing through the conductor is
I = `n eAv_d`
I = `n e A((eE)/m)τ`
Also, field E can be expressed as
E = `V/L`
The current flowing through the conductor is:
I = `(n e^2VAτ)/(mL)`
or `V/I = (mL)/(n e^2τA)`
or `R = (mL)/(n e^2τA)` ...`("from Ohm's law" V/I = R)`
or `R = m/(n e^2τ)(L/A)`
Electrical resistivity, `rho = m/(n e^2τ)` `...[∵ R = rho L/A]`
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