Question

The resistance of a wire at room temperature ${{30}^{0}}C$ is found to be $10\Omega$. Now to increase the resistance by $10\%$, the room temperature of the wire must be [The temperature coefficient of resistance of the material of the wire is 0.002 per $^{0}C$]
$\begin{aligned} & \text{A}\text{. 3}{{\text{6}}^{0}}C \\\ & \text{B}\text{. 8}{{\text{3}}^{0}}C \\\ & \text{C}\text{. 6}{{\text{3}}^{0}}C \\\ & \text{D}\text{. 3}{{\text{3}}^{0}}C \\\ \end{aligned}$

Answer 1

To solve this question we will use the concept of dependence of resistance of material on the temperature of the material. Obtain the expression to find the change in resistance of a material with the change in temperature. Put the given values to find the required answer.

Complete answer:
Resistance of an object can be defined as the measure of the opposition given by the material to the flow of current or electricity through the material. The resistance of a material depends on the temperature of the material i.e. with the change in temperature of the material, the resistance of the material will also change. Mathematically, we can express the change in resistance of the material with the temperature as,
$R={{R}_{0}}(1+\alpha \Delta T)$
Where, R is the resistivity of the material at temperature T
${{R}_{0}}$ is the initial resistance or the resistance at the initial temperature,
$\alpha$ is the temperature coefficient of resistance
The resistance of a wire at room temperature ${{30}^{0}}C$ is $10\Omega$.
$10={{R}_{0}}\left( 1+30\alpha \right)$
After we increase the resistance by $10\%$, the resistance will be,
$R=10+10\times \dfrac{10}{100}=11\Omega$
We can write,
$11={{R}_{0}}\left( 1+T\alpha \right)$
Dividing the above equation one by another, we get that,
$\begin{aligned} & \dfrac{11}{10}=\dfrac{{{R}_{0}}\left( 1+T\alpha \right)}{{{R}_{0}}\left( 1+30\alpha \right)} \\\ & 10+10\alpha T=11+330\alpha \\\ & 10\times 0.002T=1+330\times 0.002 \\\ & T=\dfrac{1+.660}{.02} \\\ & T={{83}^{0}}C \\\ \end{aligned}$
To increase the resistance by $10\%$, the room temperature of the wire must be ${{83}^{0}}C$.

The correct option is (B).

Note:
The resistivity of a material depends on the temperature of the material. The resistivity of the material increases with increase in temperature. This dependence of the resistivity on temperature can also be written as the dependence of the resistance on temperature as the above used mathematical expression.