Question

Assuming a constant temperature coefficient of resistivity, what is the maximum percent decrease in the resistance of a constantan wire starting at \({20.0°}\;{\rm{C}}\) ?

Short answer

The maximum percentage decrease in the resistance of a constantan wire is obtained as: \(0.06\;\% \).

Step-by-step solution

Define Resistance

In an electrical circuit, resistance depends on the length of the wire, the cross-sectional area of the wire, and the type of wire through which the current is flowing.

Concepts and Principles

The dependency resistance of a conductor on temperature 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 temperature coefficient of resistivity for constantan is, \(\alpha = 0.002 \times {10^{ - 3}}^\circ \;{{\rm{C}}^{ - 1}}\).
• The value\(\alpha \)is assumed to be constant.
• The initial resistance of the wire is given at temperature, \({T_0}{\rm{ }} = {\rm{ }}20.0^\circ \;{\rm{C}}\).

Evaluating maximum percentage decreases in resistance of the wire

Resistance of the constantan wire is expressed as a function of temperature change is found from the above equation as:

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

If interested in the minimum temperature of the wire, we substitute \(T = - {273.15^\circ }\;{\rm{C}}\) , which is the minimum possible temperature material can maintain.

Entering the values and we obtain:

\(\begin{align}R{\rm{ }} &= {\rm{ }}{R_0}\left( {1 + \left( {0.002 \times {{10}^{ - 3}}^\circ \;{{\rm{C}}^{ - 1}}} \right)\left( { - {{273.15}^\circ }\;{\rm{C}} - {{20.0}^\circ }\;{\rm{C}}} \right)} \right)\\R{\rm{ }} &= {\rm{ }}{R_0}\left( {1 + \left( {0.002{\rm{ }} \times {\rm{ }}{{10}^{ - 3}}^\circ \;{{\rm{C}}^{ - 1}}} \right)\left( { - {{293.15}^\circ }\;{\rm{C}}} \right)} \right)\\R{\rm{ }} &= {\rm{ }}0.9994137{R_0}\end{align}\)

The maximum percent decrease in the resistance of the wire is then obtained as:

\({\rm{Percentage}}\;{\rm{decrease}} = {\rm{ }}\left( {\frac{{{R_0} - R}}{{{R_0}}}} \right)\left( {100{\rm{ }}\% } \right)\)

Entering the values and we obtain:

\(\begin{align}{\rm{Percentage}}\;{\rm{decrease}} &= {\rm{ }}\left( {\frac{{{R_0} - 0.9994137{\rm{ }}{R_0}}}{{{R_0}}}} \right)\left( {100{\rm{ }}\% } \right)\\ &= {\rm{ }}0.06\;\% \end{align}\)

Therefore, the maximum percentage decrease in the resistance of a constantan wire is: \(0.06{\rm{ }}\% \).