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Question
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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Solution
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.
RELATED QUESTIONS
Define the term drift velocity.
Write its (‘mobility’ of charge carriers) S.I. unit
Derive an expression for drift velocity of free electrons in a conductor in terms of relaxation time.
When a current is established in a wire, the free electrons drift in the direction opposite to the current. Does the number of free electrons in the wire continuously decrease?
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 relaxation time τ is nearly independent of applied E field whereas it changes significantly with temperature T. First fact is (in part) responsible for Ohm’s law whereas the second fact leads to variation of ρ with temperature. Elaborate why?
Explain how free electrons in a metal at constant temperature attain an average velocity under the action of an electric field. Hence, obtain an expression for it.
A potential difference (V) is applied across a conductor of length 'L' and cross-sectional area 'A'.
How will the drift velocity of electrons and the current density be affected if another identical conductor of the same material were connected in series with the first conductor? Justify your answers.
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.
