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प्रश्न
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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उत्तर
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]`
संबंधित प्रश्न
Derive an expression for drift velocity of free electrons.
Write its (‘mobility’ of charge carriers) S.I. unit
Estimate the average drift speed of conduction electrons in a copper wire of cross-sectional area 2.5 × 10−7 m2 carrying a current of 1.8 A. Assume the density of conduction electrons to be 9 × 1028 m−3.
A conductor of length ‘l’ is connected to a dc source of potential ‘V’. If the length of the conductor is tripled by gradually stretching it, keeping ‘V’ constant, how will (i) drift speed of electrons and (ii) resistance of the conductor be affected? Justify your answer.
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 ______.
The drift velocity of a free electron inside a conductor is ______
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?
Define relaxation time.
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.
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.
