A carbon resistor is to be used as a thermometer. On a...

A carbon resistor is to be used as a thermometer. On a winter day when the temperature is 4.0$^\circ$C, the resistance of the carbon resistor is 217.3 $\Omega$. What is the temperature on a spring day when the resistance is 215.8 $\Omega$? (Take the reference temperature $T{_0}$ to be 4.0$^\circ$C.)

Key Concepts

Temperature Coefficient of Resistivity

This concept quantifies how much a material's electrical resistivity changes with temperature. It is defined as the fractional change in resistance per degree change in temperature and indicates whether the material’s resistance increases or decreases as temperature varies. A negative value means that the resistance decreases when the temperature increases.

Linear Temperature Dependence of Resistance

In many materials, especially over a limited temperature range, the resistance varies linearly with temperature. This relationship is typically modeled with the equation R = R?(1 + ?(T - T?)), where R? is the resistance at a reference temperature T? and ? is the temperature coefficient. This linear approximation is essential for converting changes in resistance into corresponding temperature changes.

Temperature Measurement Using Resistive Sensors

Resistive temperature sensors, such as those using carbon resistors, exploit the predictable change in resistance with temperature to measure thermal variations. By calibrating the sensor at a known temperature (the reference point), the sensor can accurately determine unknown temperatures through observed changes in resistance. This method forms the basis for various thermometry applications where electrical resistance is a measurable proxy for temperature.

Transcript

00:01 So we're given that a carbon resistor is going to be used as a thermometer.

00:10 And on a particular day, there is a temperature at 4 degrees celsius, which has a corresponding resistance of 217 .3 oms.

00:26 And i've just labeled these with w because the problem sounds on a winter day.

00:30 So just put w for winter and we're told that the reference temperature is 4 degrees celsius and they want to know on a different day when there is a when the resistor has a resistance of 215 .8 oms what is the temperature on that day so we can solve this problem if we have an equation that relates resistance to temperature, which we actually have one.

01:06 So equation 25 .12 gives you the resistance is a function of temperature is equal to some reference resistance, one plus alpha, which is some coefficient, and then the temperature.

01:32 And alpha can be found in table 25 point two.

01:40 So and this alpha will be alpha corresponding to carbon.

01:48 And it has units of inverse celsius.

01:56 So our goal here is to find r of t.

02:01 The resistance is a function of temperature, but we don't know this reference resistance.

02:10 So we know alpha, which we can find from the table and we know these two temperatures.

02:18 So using the information that we're given, let's find r of t, or rather let's find r not, because we have an r of t.

02:37 Okay, so at a particular temperature of 4 degrees celsius, we have rw, so that's equal to r0, 1 plus alpha t w minus t not but the reference temperature in d w are the same as we can see here right so this difference is zero so that tells us that are the reference the reference resistance is equal to r w which is already given to us as 217 .3 oms 7 .3 oms 7 .3...

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