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प्रश्न
Define relaxation time of the free electrons drifting in a conductor. How is it related to the drift velocity of free electrons? Use this relation to deduce the expression for the electrical resistivity of the material.
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उत्तर
Relaxation time (τ), it is the short time for which a free electron accelerates before it undergoes a collision with the positive ion in the conductor. Or, we can say it is the average time elapsed between two successive collisions. It is of the order 10−14 s. It decreases with increase of temperature and is given as
`vecV_d = vecatau`
`or vecV_d = (-eE)/m tau [because veca =-(evecE)/m]`
Where `vecV_d` is the drift velocity E is the applied electric field. e and m are the charge and mass of electron respectively.
Again consider the conductor with length l and A as area of cross-section. Let n be the number of electrons per unit volume in the conductor.
`V_d = -(eE)/m tau`(Magnitude of drift velocity)
The current flowing through the conductor due to drift
I = nAvde
Substituting value of νd
`I = nA ((eEtau)/m)e`
`I = (nAe^2Etau)/m`
If V is potential difference applied across the two ends then
`E = V/l`put in above equation
`So I = (nAe^2Vtau)/(ml)`
`V/I = (ml)/("ne"^2tauA)`
Now, According to ohm’s law `V/1 = R`(Resistance of conductor)
Thus,
`R = m/("ne"^2tau) l/A`
Compare this with formula of resistance `R =rho*l/A`
Where ρ is the resistivity of the material we get
`rho = m/("ne"^2tau)`
Thus electrical resistivity depends inversely on the relaxation time τ.
संबंधित प्रश्न
Define the term drift velocity.
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