Question

(a) Of what material is a wire made, if it is\(25.0{\rm{ }}m\)long with a\(0.100{\rm{ }}mm\)diameter and has a resistance of\(77.7{\rm{ }}\Omega \)at ? (b) What is its resistance at ?

Short answer

1. The material from which the wire is made is gold.
2. The resistance of the wire at temperature is obtained as: \(R = {\rm{110}}\;{\rm{\Omega }}\).

Step-by-step solution

Define Resistance

In an electrical circuit, resistance is a measure of the resistance to current flow. The Greek letter omega\({\rm{(\Omega )}}\)is used to represent resistance in ohms.

Concepts and Principles

The electrical resistance of the value\(R\)of a resistive circuit element depends on its geometric structure (its length symbolized as\(L\)and cross-sectional area symbolised as\(A\))and on the resistivity\(\rho \)of the material of which it is made:

\({R_0}{\rm{ }} = {\rm{ }}{\rho _0}\frac{L}{A}\)………………(I)

The resistance of a conductor varies approximately linearly with temperature according to the expression given as:

\(R{\rm{ }} = {\rm{ }}{R_0}{\rm{ }}(1{\rm{ }} + {\rm{ }}\alpha {\rm{ }}(T{\rm{ }} - {\rm{ }}{T_0}))\)................(II)

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 length of the wire is:\(L{\rm{ }} = {\rm{ }}25.0{\rm{ }}m\).
• Diameter of the wire is:

\(\begin{aligned}{c}D{\rm{ }} &= {\rm{ }}(0.100{\rm{ }}mm)(\frac{{1{\rm{ }}m}}{{1000{\rm{ }}mm}})\\ &= {\rm{ }}0.100{\rm{ }} \times {\rm{ }}{10^{ - 3}}{\rm{ }}m\end{aligned}\)

• The r, The radius of the wire, is :

\(\begin{aligned}{c}r{\rm{ }} &= {\rm{ }}D/2{\rm{ }}\\ &= {\rm{ }}0.05{\rm{ }}x{\rm{ }}{10^{ - 3}}{\rm{ }}m\end{aligned}\)

• Resistance of the wire at the temperature\({T_0}{\rm{ }} = {\rm{ }}20.0^\circ {\rm{ }}C\)is:

\({R_0}{\rm{ }} = {\rm{ }}77.7{\rm{ }}\Omega \)

Finding material from which wire is made

1. Resistance of the wire at temperature\({{\rm{T}}_{\rm{0}}}\)is found from the above equation as:\({R_0}{\rm{ }} = {\rm{ }}{\rho _0}\frac{L}{A}\)

The value ofis the cross-sectional area of the wire:

Solving for\({\rho _0}\):

Entering the values and we obtain:

The coefficient of the resistivity is related to the material known as gold.

Evaluating the resistance of the wire at the temperature

Resistance of the wire is expressed as a function of temperature from the second equation as:

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

The temperature coefficient of resistivity of gold is:\(\alpha {\rm{ }} = {\rm{ }}3.4{\rm{ }} \times {\rm{ }}{10^{ - 3{\rm{ }}o}}{\rm{ }}{C^{ - 1}}\).

Entering the values and we obtain:

\(\begin{aligned}{c}R{\rm{ }} &= {\rm{ }}(77.7{\rm{ }}\Omega )(1{\rm{ }} + {\rm{ }}(3.4{\rm{ }} \times {\rm{ }}{10^{ - 3{\rm{ }}o}}{\rm{ }}{C^{ - 1}})({150^o}{\rm{ }}C{\rm{ }} - {\rm{ }}{20^o}{\rm{ }}C)\\ &= {\rm{ }}110{\rm{ }}\Omega \end{aligned}\)

The resistance is\({\rm{110}}\;{\rm{\Omega }}\)instead of\({\rm{112}}\;{\rm{\Omega }}\)as it is rounded to two significant figures.

Therefore, we get:

1. The material from which wire is made is gold.
2. Resistance of wire at temperature is:\(R = {\rm{110}}\;{\rm{\Omega }}\).