Question

A silver wire has a resistance of $2.1\Omega$ at ${27.5^0}C$ and a resistance of $2.7\Omega$ at ${100^0}C$ Determine the temperature coefficient of resistivity of silver.

Answer 1

Hint- In order to solve this question, we will use the formula for calculating the change in the resistivity of the metal with increase in the temperature. We will create two equations using the formula and then solve for the coefficient of equations and then proceed further.

Complete step-by-step answer:
Formula used- $R = {R_0}\left( {1 + \alpha T} \right)$

Where R is the resistance at temperature T

$\alpha$ is the coefficient of resistivity
And ${R_0}$ is the resistance at room temperature.

Given

Temperature ${T_1} = {27.5^0}C$
Resistance of the silver wire at temperature ${T_1}$ is ${R_1} = 2.1\Omega$
Temperature ${T_2} = {100^0}C$

Resistance of the silver wire at temperature ${T_2}$ is ${R_2} = 2.7\Omega$

Let $\alpha$ be the coefficient of resistivity.

As we know that $R = {R_0}\left( {1 + \alpha T} \right)$

Substituting the value of resistance and temperature at different times in the above equation

${R_1} = {R_0}\left( {1 + \alpha {T_1}} \right)...............\left( 1 \right)$
${R_2} = {R_0}\left( {1 + \alpha {T_2}} \right)..................\left( 2 \right)$

Dividing equation (1) and (2), and solving for $\alpha$ we get

$\alpha = \dfrac{{{R_2} - {R_1}}}{{{R_1}\left( {{T_2} - {T_1}} \right)}}$

Substituting the value of resistance and temperature from the given conditions

$\alpha = \dfrac{{2.7 - 2.1}}{{2.1\left( {100 - 27.5} \right)}} \\\ \alpha = {0.0039^0}{C^{ - 1}} \\\$

Hence, the coefficient of resistivity for silver is $\alpha = {0.0039^0}{C^{ - 1}}$

Note- For conductor resistance values at any temperature other than the normal temperature (usually defined at 20 Celsius) on the basic resistance table, another formula must be used to determine: Conductor values at some other temperature than the normal temperature (usually defined at 20 Celsius) on the basic resistance table must be calculated using another calculation. The coefficient is a positive number for pure metals, indicating the resistance decreases as the temperature increases. The coefficient is a negative number for the elements carbon, silicon and germanium, indicating that the resistance reduces with rising temperature