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2015-07-30T01:51:28+05:30

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We will find the relation for small temperature changes.
       resistance  R = ρ L / A

coefficient of linear expansion = α
length of conductor:  L = L₀ ( 1 + α ΔT)               ΔL = α L₀  ΔT
β = coefficient of expansion in area : = 2 α
Area of cross section:    A =  A₀ (1+ 2 α ΔT)        ΔA = 2α A₀ ΔT

Resistivity    ρ = ρ₀ (1 + Aρ  ΔT)           Δρ =  ρ₀  Aρ  ΔT
Resistance   R = R₀  (1 + Ar  ΔT)      :  ΔR = Ar  R₀  ΔT

If  ΔA = 2 α ΔT  is very small then,  and  for small ΔT,

R₀ = ρ₀ L₀ / A₀

R = ρ₀ (1 + Aρ ΔT)  L₀ (1 + α ΔT) / [ A₀ (1 + 2 α ΔT) ]
   = (ρ₀ L₀ / A₀) (1 + Aρ ΔT) (1 + α ΔT) (1 - 2 α ΔT)
   = R₀  (1 + Aρ ΔT) (1 - α ΔT)              ignoring the 2 α² ΔT²   term
   =  R₀ [ 1 + (Aρ - α) ΔT ]                    ignoring  the  Aρ α ΔT²  term

Ar = (Aρ - α)

If the conductor does not expand, then the value of coefficient of resistance is same as the coefficient of resistivity.  But  as the area of cross section increases with temperature, the coefficient decreases by  α.


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