Question

A length of aluminum wire is connected to a precision 10.00-V power supply, and a current of 0.4076 A is precisely measured at 30.0∘C. The wire is placed in a new environment of unknown temperature where the measured current is 0.3502 A. The temperature coefficient of resistivity of aluminum is 0.00429. What is the unknown temperature?

Answer 1

Answer:

$68.3^{\circ}C$

Explanation:

The relationship between voltage, resistance and current in a metal wire is given by

$V=RI$

where

V is the voltage

R is the resistance

I is the current

Moreover, the resistance of a metal wire as a function of the temperature is:

$R(T)=R_0(1+\alpha(T-T_0))$ (1)

where

$R_0$ is the resistance at temperature $T_0$

$\alpha$ is the temperature coefficient

In this problem, at 30.0∘C, we have

V = 10.00 V

I = 0.4076 A

So the resistance at this temperature is

$R_0=\frac{V_0}{I_0}=\frac{10.00}{0.4076}=24.53 \Omega$

Then, at the unknown temperature T, the current is

I = 0.3502 A

So the resistance is

$R(T)=\frac{V}{I}=\frac{10.00}{0.3502}=28.56\Omega$

Substituting and solving for (1), we find the unknown temperature:

$T=T_0+\frac{\frac{R}{R_0}-1}{\alpha}=30.0+\frac{\frac{28.56}{24.53}-1}{0.00429}=68.3^{\circ}C$