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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
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.
On the basis of electron drift, derive an expression for resistivity of a conductor in terms of number density of free electrons and relaxation time. On what factors does resistivity of a conductor depend?
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.
Derive an expression for drift velocity of free electrons in a conductor in terms of relaxation time.
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 _______________ .
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 .
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.
