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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
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 2·7 A. Assume the density of conduction electrons to be 9 × 1028 m−3
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?
Obtain the expression for the current flowing through a conductor having number density of the electron n, area of cross-section A in terms of the drift velocity vd .
Amount of charge in coulomb required to deposit one gram equivalent of substance by electrolysis is:-
Is the momentum conserved when charge crosses a junction in an electric circuit? Why or why not?
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.
The potential difference applied across a given conductor is doubled. How will this affect (i) the mobility of electrons and (ii) the current density in the conductor? Justify your answers.
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.
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.
