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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.
Estimate the average drift speed of conduction electrons in a copper wire of cross-sectional area 1.0 × 10−7 m2 carrying a current of 1.5 A. Assume the density of conduction electrons to be 9 × 1028 m−3
The number density of free electrons in a copper conductor is 8.5 × 1028 m−3. How long does an electron take to drift from one end of a wire 3.0 m long to its other end? The area of cross-section of the wire is 2.0 × 10−6 m2 and it is carrying a current of 3.0 A.
(a) drift speed
(b) current density
(c) electric current
(d) electric field
Why alloys like constantan and manganin are used for making standard resistors?
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 drift velocity of a free electron inside a conductor is ______
- Consider circuit in figure. How much energy is absorbed by electrons from the initial state of no current (ignore thermal motion) to the state of drift velocity?
- Electrons give up energy at the rate of RI2 per second to the thermal energy. What time scale would one associate with energy in problem (a)? n = no of electron/volume = 1029/m3, length of circuit = 10 cm, cross-section = A = (1mm)2
Derive an expression for resistivity of a conductor in terms of the number density of charge carriers in the conductor and relaxation time.
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.
