Question

Resistance of tungsten wire at $150^\circ C$ is $133\Omega $. Its resistance temperature coefficient is $0.0045/^\circ C$. The resistance of this wire at $500^\circ C$ will be?
A. $180\Omega $
B. $225\Omega $
C. $258.8\Omega $
D. $317\Omega $

Answer 1

Hint:In order to solve this you have to remember the formula for resistance of a resistor at any given temperature and the resistance temperature coefficient is given in the question which is a constant and has the same value for the same type of wire.

Formula used:
$\dfrac{{{R_t}}}{{{R_t}^\prime }} = \dfrac{{1 + \alpha t}}{{1 + \alpha t'}}$
Where \[{R_t}\] and ${R_t}^\prime $ are the resistance of a resistor at temperature $t$ and $t'$ respectively and $\alpha $ is the resistance temperature coefficient.

Complete step by step solution:
Here in this question resistance of tungsten wire is given as ${R_t} = 133\Omega $ at the temperature $t = 150^\circ C$ .
And also given the temperature coefficient is given as $\alpha = 0.0045/^\circ C$
We have to find the resistance of that tungsten wire at $t' = 500^\circ C$
By using the formula, we have
$\dfrac{{{R_t}}}{{{R_t}^\prime }} = \dfrac{{1 + \alpha t}}{{1 + \alpha t'}}$
Where \[{R_t}\] and ${R_t}^\prime $ are the resistance of a resistor at temperature $t$ and $t'$ respectively and $\alpha $ is the resistance temperature coefficient.
On putting all the values, we have
$\dfrac{{133}}{{{R_t}^\prime }} = \dfrac{{1 + 0.0045\left( {150} \right)}}{{1 + 0.0045(500)}}$
On further solving, we have
$\dfrac{{133}}{{{R_t}^\prime }} = \dfrac{{1.67}}{{3.25}}$
From the above equation, on further solving we have
${R_t}^\prime = \dfrac{{432.25}}{{1.67}} = 258.8\Omega $
The resistance of this tungsten wire at $500^\circ C$ is $258.8\Omega $

Therefore, the correct option is (C) $258.8\Omega $

Additional information:
The electric resistance due to temperature depends on
-the temperature rise
-resistance temperature coefficient
-the value of resistance at an initial temperature.
There are two types of resistance temperature coefficient
-Positive resistance temperature coefficient
-Negative resistance temperature coefficient

Note:The resistance and the electrical resistivity of the material is affected by the temperature. Basically, the temperature coefficient of resistance is defined as the change in electric resistance with respect to per degree change in temperature. The value of resistance temperature coefficient can vary depending on the type of material.