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A metallic wire has a resistance of 3.0Ω at 0℃ and 4.8Ω at 150℃. Find the temperature coefficient of resistance of its material.

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Last updated date: 15th Jul 2024
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Answer
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Hint: The change in resistance per unit reference resistance is directly proportional to the change in temperature of the material, and the proportionality constant of this relation is called the temperature coefficient of resistance.

Complete step by step answer:
In this question there is a metallic wire which has a resistance of 3 ohms at temperature of 0℃, and a resistance of 4.8 ohms at a temperature of 150℃. We must find out the temperature coefficient of resistance of its material. But before that, we must understand how the resistance of a material depends on its temperature.
We know how a material offers electrical resistance, i.e. when the free electrons inside the metal Have a low drift speed. The drift speed of the free electrons can decrease if the material has a very list number of free electrons, or if the free electrons get too excited (due to rise in temperature) and have too many collisions among themselves. This is the reason how the temperature rise causes a rise in the resistance of the material.
Now that we know how the resistance changes with temperature, we must calculate the coefficient with which we can relate those changes. We have seen that the rise in temperature causes a rise in the resistance and vice versa. Experimentally, the relation that is found is
${R_T} = {R_0}\left( {1 + \alpha \left( {T - 0} \right)} \right)$
Where RT = Resistance at temperature T℃
R0 = Resistance at temperature 0℃
α = Temperature coefficient of resistance
T = Temperature 0 = reference temperature
On removing the proportionality, we get a proportionality constant which is called the temperature coefficient of resistance.

 $ {R_T} = {R_0}\left( {1 + \alpha T} \right) \\
  \dfrac{{{R_T}}}{{{R_0}}} = 1 + \alpha T \\
  \dfrac{{{R_T} - {R_0}}}{{{R_0}}} = \alpha T \\
  \dfrac{{{R_T} - {R_0}}}{{{R_0}T}} = \alpha = \dfrac{{4.8 - 3}}{{3 \times 150}} = 0.004^\circ {C^{ - 1}} \\ $

Note:
It looks like that the temperature is directly proportional to the resistance of the material, but the actual relation is what is mentioned above which is not direct proportionality.The temperature rise causes a rise in the resistance of the material.