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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
Derive an expression for drift velocity of free electrons.
When electrons drift in a metal from lower to higher potential, does it mean that all the free electrons of the metal are moving in the same direction?
Consider the following statements.
(A) Free-electron density is different in different metals.
(B) Free-electron density in a metal depends on temperature.
Seebeck Effect is caused _____________ .
The position-time relation of a particle moving along the x-axis is given by x = a - bt + ct2 where a, band c are positive numbers. The velocity-time graph of the particle is ______.
When a current I is set up in a wire of radius r, the drift velocity is vd· If the same current is set up through a wire of radius 2 r, the drift velocity will be:
Amount of charge in coulomb required to deposit one gram equivalent of substance by electrolysis is:-
The identical conductors maintained at same temperature are given potential difference in the ratio 1 : 2. Then the ratio of their drift velocities is ______.
Two conductors, made of the same material have equal lengths but different cross-sectional areas A1 and A2 (A1 > A2). They are connected in parallel across a cell. Show that the drift velocities of electrons in two conductors are equal.
The drift velocity of electrons in a conductor connected to a battery is given by vd = `(−"eE" τ)/"m"`. Here, e is the charge of the electron, E is the electric field, τ is the average time between collisions and m is the mass of the electron.
Based on this, answer the following:
- How does the drift velocity change with a change in the potential difference across the conductor?
- A copper wire of length 'l' is connected to a source. If the copper wire is replaced by another copper wire of the same area of cross-section but of length '4l', how will the drift velocity change? Explain your answer.
