Question

Of what material is a resistor made if its resistance is \({\bf{40}}{\bf{.0}}\;{\bf{\% }}\)greater at than at \({\bf{20}}{\bf{.0°}}\;{\bf{C}}\)?

Short answer

The wire is made from the material known as Iron.

Step-by-step solution

Define Resistance

In an electrical circuit, the flow of current varies with the variation in resistance

In an electrical circuit, the flow of current varies with the variation in resistance.

The resistance of a conductor can be given as:

\(R = {R_0}\left( {1 + \alpha \left( {T - {T_0}} \right)} \right)\)

The value of \({R_0}\) is the resistance at some reference temperature, and the value of \({T_0}\) and the value of \(\alpha \) is the temperature coefficient of resistivity.

The given data

The initial temperature of the wire is:\({T_0} = 20^\circ \;{\rm{C}}\).

The resistance of the wire is forty percent greater at \(T = {100^\circ }\;{\rm{C}}\) than at \({T_0}\).

Evaluating the material

The resistance of the wire is expressed as a function of temperature change by the equation as:

\(R = {R_0}\left( {1 + \alpha \left( {T - {T_0}} \right)} \right)\) (1)

As the resistance of the wire is forty percent at temperature\(T\)than at temperature\({T_0}\), the resistance of the wire will be:

\(\begin{align}R{\rm{ }} &= {\rm{ }}{R_0} + 40\;\% \left( {{R_0}{\rm{ }}} \right)\\ &= {\rm{ }}1.40{R_0}\end{align}\)

Substitute the values in equation 1, and we get:

\(\begin{align}1.40{R_0}{\rm{ }} &= {\rm{ }}{R_0}\left( {1 + \alpha \left( {T - {T_0}} \right)} \right)\\1.40{\rm{ }} &= {\rm{ }}1 + \alpha \left( {T - {T_0}} \right)\end{align}\)

Rearranging and solving for\(\alpha \):

\(\begin{align}1.40 - 1{\rm{ }} &= {\rm{ }}\alpha \left( {T - {T_0}} \right)\\0.40{\rm{ }} &= {\rm{ }}\alpha \left( {T - {T_0}} \right)\\\alpha {\rm{ }} &= {\rm{ }}\frac{{0.40}}{{T - {T_0}}}\end{align}\)

Entering the values for\(T\)and\({T_0}\), we obtain:

\(\begin{align}\alpha {\rm{ }} &= {\rm{ }}\frac{{0.40}}{{{{100}^\circ }\;{\rm{C }} - {{20}^\circ }\;{\rm{C}}}}\\ &= {\rm{ }}5.00 \times {10^{ - 3}}^\circ \;{{\rm{C}}^{ - 1}}\end{align}\)

We do observe that this coefficient of resistivity is related to iron.

Therefore, the wire is made of Iron.