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### College Physics (Urone)

Book edition 1st Edition
Author(s) Paul Peter Urone
Pages 1272 pages
ISBN 9781938168000

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

The wire is made from the material known as Iron.

See the step by step solution

## Step 1: 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.

## Step 3: 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 $${T_0}$$.

## Step 4: 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.