Question

At \[{\text{100}}\,^\circ {\text{C}}\] , the resistance of a conducting coil is \[{\text{4}}{\text{.2}}\,\Omega \] . If the temperature coefficient of resistance is \[{\text{0}}{\text{.004}}\,^\circ {\text{C}}\] , what will be the resistance of \[{\text{0}}\,^\circ {\text{C}}\] ? (in ohm)
A. \[{\text{3}}\]
B. \[{\text{5}}\]
C. \[{\text{4}}\]
D. \[{\text{3}}{\text{.5}}\]

Answer 1

Hint: First of all, we will find out the difference in temperature. Then we will apply the formula which gives the relation between resistances at different temperatures linked with coefficient of resistance. We will substitute the required values and manipulate accordingly to obtain the result.

Complete step by step answer:
In the given problem, we are supplied with the following data:
The initial temperature is \[{\text{100}}\,^\circ {\text{C}}\] .
The resistance of a conducting coil at \[{\text{100}}\,^\circ {\text{C}}\] is \[{\text{4}}{\text{.2}}\,\Omega \] .
The temperature coefficient of resistance is \[{\text{0}}{\text{.004}}\,^\circ {\text{C}}\] .
We are asked to find the resistance at temperature \[{\text{0}}\,^\circ {\text{C}}\] .

To begin with, we will first find the change in temperature, which is calculated as:
\[\Delta T = {T_1} - {T_2}\] …… (1)
Where,
\[\Delta T\] indicates the change in temperature.
\[{T_1}\] indicates the initial temperature.
\[{T_2}\] indicates the final temperature.

Now, substituting the required values in the equation (1), we get:
\[
\Delta T = 100\,^\circ {\text{C}} - 0\,^\circ {\text{C}} \\
\Rightarrow \Delta T = 100\,^\circ {\text{C}} \\
\]
We have a formula which gives resistance at any temperature, which is linked to the coefficient of resistance as given below:
\[{R_{\text{T}}} = {R_0}\left( {1 + \alpha \Delta T} \right)\] …… (1)
Where,
\[{R_{\text{T}}}\] indicates the resistance at temperature \[T\] .
\[{R_0}\] indicates the resistance at temperature \[{\text{0}}\,^\circ {\text{C}}\] .
\[\alpha \] indicates coefficient of resistance.
\[\Delta T\] indicates the change in temperature.

From equation (1), we get after rearranging the equation:
\[\dfrac{{{R_{\text{T}}}}}{{\left( {1 + \alpha \Delta T} \right)}} = {R_0}\] …… (2)

Now, we substitute the required values in equation (2) and manipulate:
\[
{R_0} = \dfrac{{{R_{\text{T}}}}}{{\left( {1 + \alpha \Delta T} \right)}} \\
\Rightarrow {R_0} = \dfrac{{4.2}}{{1 + {\text{0}}{\text{.004}} \times 100}} \\
\Rightarrow {R_0} = \dfrac{{4.2}}{{1.4}} \\
\Rightarrow {R_0} = 3\,\Omega \\
\]
Hence, the resistance of \[{\text{0}}\,^\circ {\text{C}}\] will be \[3\,\Omega \] .
The correct option is A.

Additional information: In an electrical circuit, resistance is a measure of the opposition to current flow. Resistance, signified by the Greek letter omega, is measured in ohms.
The resistance-change component is called the resistance temperature coefficient, per degree Celsius of temperature change. For an increase in temperature, a positive coefficient for a substance means that its resistance increases.

Note:While solving the problem, we should keep in mind that at higher temperature the resistance of the material tends to increase. It allows atoms to vibrate further by heating the metal conductor, which in turn makes it more difficult for the electrons to flow, raising the resistance.