Question

At what temperature will the resistance of a copper wire become three times its value at \[{{0}^{0}}C\]? (Temperature coefficient of resistance of copper is\[4\times {{10}^{-3}}\Omega \text{ per}{{\text{ }}^{0}}C\])
\[\begin{align}
  & \text{A) 40}{{\text{0}}^{0}}C \\
 & \text{B) 45}{{\text{0}}^{0}}C \\
 & \text{C) 50}{{\text{0}}^{0}}C \\
 & \text{D) 60}{{\text{0}}^{0}}C \\
\end{align}\]

Answer 1

Hint: We are given that the resistance of the copper wire at some temperature becomes thrice the value of resistance at \[{{0}^{0}}C\]. We know the temperature coefficient of the copper wire also. From all these, we can easily find the unknown temperature.

Complete answer:
We know that the resistance of any conductor is directly related to its temperature. Any rise in temperature results in significant rise in the resistance of the wire. We know the formula which connects these relations. It is related by the temperature coefficient of resistance, which is a constant for any material. The formula is given by –
\[R={{R}_{0}}(1+\alpha T)\]
Where, R is the resistance at temperature T,
\[{{R}_{0}}\] is the resistance of the material at \[{{0}^{0}}C\],
\[\alpha \]is the temperature coefficient of resistance of the material.
Now, let us consider our situation. T is given that a copper has a resistance at some temperature ‘T’ such that it is three times the resistance of the copper wire at \[{{0}^{0}}C\]. So, we have, the following data for our formula to be used –
The resistance at Temperature T,
\[R=3{{R}_{0}}\]
The temperature coefficient of resistance of copper –
\[\alpha =4\times {{10}^{-3}}\Omega \text{ per}{{\text{ }}^{0}}C\]
Now, let us equate the above values in the formula as –
\[\begin{align}
  & R={{R}_{0}}(1+\alpha T) \\
 & \Rightarrow 3{{R}_{0}}={{R}_{0}}(1+4\times {{10}^{-3}}T) \\
 & \Rightarrow 3=1+4\times {{10}^{-3}}T \\
 & \Rightarrow T=\dfrac{2}{4\times {{10}^{-3}}}={{500}^{0}}C \\
\end{align}\]
The required temperature is \[{{500}^{0}}C\].

The correct answer is option C.

Note:
The temperature coefficient of a material describes what type of substance – conductor or semiconductor – it is. The metal conductors have a positive coefficient, whereas the semiconductors have it negative, i.e, With rise in temperature the resistance of conductors increases, but decreases for a semiconductor.