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What is temperature co-efficient of resistance? Derive an expression for temperature co-efficient of resistance.
Ans.
The change of resistance of a material with temperature is expressed by means of temperature co-efficient of resistance.
Let a conductor having initial resistance Rₒ at 0 ⁰C and Rᵼ be the final resistance at t ⁰C.
So increase in temperature……………
ΔR = Rt- Rₒ
This increase in resistance is…………
(a) directly proportional to its initial resistance i.e., Rt- Rₒ ∝ Rₒ
(b) directly proportional to the rise in temperature i.e., Rt- Rₒ ∝ t
(c ) depends upon the nature of the material.
Combining first two we get,
Rt- Rₒ ∝ Rₒt
Rt- Rₒ = αₒRₒt ……………………….. (i)
Where αₒ is a constant & is called temperature co-efficient of resistance at 0 ⁰C. Its value depends upon the nature of material and temperature.
Rearranging equation ( i ), we get,
Rt = Rₒ + αₒRₒt
= Rₒ( 1 + αₒt) …………………. (ii)
So temperature co-efficient of resistance is expressed as given below from equation (i)..........
αₒ = Rt- Rₒ/ Rₒt
= Increase in resistance/ohm original resistance/⁰C rise in temperature
Hence, temperature co-efficient of resistance of a material is the increase in resistance per ohm original resistance per ⁰C rise in temperature.
If a conductor has a resistance Rₒ, R₁ & R₂ at 0 ⁰C, t₁ ⁰C & t₂ ⁰C respectively, then…..
R₁ - Rₒ = α Rₒ t₁
R₁ = Rₒ + α Rₒ t₁
R₁ = Rₒ(1 + αt₁) ……………. (iii)
R₂ - Rₒ = α Rₒ t₂
R₂ = Rₒ + α Rₒ t₂
R₂ = Rₒ(1 + αt₂) …………..(iv)
Dividing equation (iv) by equation (iii), we get….
R₂/R₁ = Rₒ(1 + αt₂)/ Rₒ(1 + αt₁)
R₂/R₁ = (1 + αt₂)/(1 + αt₁)
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