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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 1.0 × 10−7 m2 carrying a current of 1.5 A. Assume the density of conduction electrons to be 9 × 1028 m−3
How does drift velocity of electrons in a metallic conductor vary with increase in temperature? Explain.
(a) drift speed
(b) current density
(c) electric current
(d) electric field
Consider a wire of length 4 m and cross-sectional area 1 mm2 carrying a current of 2 A. If each cubic metre of the material contains 1029 free electrons, find the average time taken by an electron to cross the length of the wire.
Consider the following statements.
(A) Free-electron density is different in different metals.
(B) Free-electron density in a metal depends on temperature.
Thomson Effect is caused _______________ .
Amount of charge in coulomb required to deposit one gram equivalent of substance by electrolysis 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
Consider two conducting wires A and B of the same diameter but made of different materials joined in series across a battery. The number density of electrons in A is 1.5 times that in B. Find the ratio of the drift velocity of electrons in wire A to that in wire B.
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.
