Problem 20 - Physics

Question

A digital thermometer employs a thermistor as the temperature-sensing element. A thermistor is a kind of semiconductor and has a large negative temperature coefficient of resistivity \(\alpha\). Suppose \(\alpha=-0.060\left(\mathrm{C}^{\circ}\right)^{-1}\) for the thermistor in a digital thermometer used to measure the temperature of a patient. The resistance of the thermistor decreases to \(85 \%\) of its value at the normal body temperature of \(37.0^{\circ} \mathrm{C}\). What is the patient's temperature?

Step-by-Step Solution

Answer

The patient's temperature is 39.5°C.

Step 1: Understand the Problem

We are given that the resistance of the thermistor decreases to 85% of its value at a normal body temperature of 37°C. The temperature coefficient of resistivity, \( \alpha \), is \( -0.060 \, (\mathrm{C}^{\circ})^{-1} \). We need to find the patient's temperature.

Step 2: Use Resistivity Coefficient Formula

The resistance of the thermistor changes according to the formula: \( R = R_0 (1 + \alpha (T - T_0)) \), where \( R_0 \) is the resistance at the reference temperature \( T_0 = 37^{\circ} \mathrm{C} \), and \( R \) is the resistance at the unknown temperature \( T \).

Step 3: Set Up the Equation

At the patient's temperature, the resistance is 85% of its value at 37°C, so \( R = 0.85 R_0 \). Plug this into the formula: \[ 0.85 R_0 = R_0 (1 + \alpha (T - 37)) \]

Step 4: Simplify the Equation

Divide both sides of the equation by \( R_0 \):\[ 0.85 = 1 + (-0.060)(T - 37) \]

Step 5: Solve for Temperature \( T \)

Rearrange and solve for \( T \):\[ 0.85 - 1 = -0.060(T - 37) \]\[ -0.15 = -0.060(T - 37) \]Divide both sides by \(-0.060\):\[ T - 37 = \frac{-0.15}{-0.060} \]\[ T - 37 = 2.5 \]Therefore, \( T = 37 + 2.5 = 39.5^{\circ} \mathrm{C} \).

Key Concepts

Temperature Coefficient of Resistivity

The temperature coefficient of resistivity, often denoted as \( \alpha \), is a crucial parameter when dealing with materials like thermistors. It describes how the resistance of a material changes in response to temperature variations. A negative temperature coefficient of resistivity means the material's resistance decreases as the temperature increases.

This behavior is typical for semiconductors like thermistors. In the given exercise, the thermistor has \( \alpha = -0.060 (\mathrm{C}^{\circ})^{-1} \). This large negative value indicates a significant change in resistance with relatively small temperature shifts.

When solving calculations involving the temperature coefficient, use the formula:
\[ R = R_0 (1 + \alpha (T - T_0)) \]
Here, \( R \) represents the resistance at an unknown temperature \( T \), \( R_0 \) is the resistance at a reference temperature \( T_0 \), and \( \alpha \) is the temperature coefficient.

This formula helps predict resistance changes, aiding in determining unknown temperatures when certain resistance changes are observed.

Semiconductor

Semiconductors are materials with a conductivity level between that of insulators and conductors. They have unique properties that allow them to be highly sensitive to temperature changes. This sensitivity is capitalized in devices like thermistors used in digital thermometers.

Thermistors are a type of semiconductor where resistance decreases significantly as temperature rises, due to the liberation of charge carriers at higher temperatures. This behaviour is represented by their negative temperature coefficient of resistivity.

Some key points about semiconductors include:

• Composed mostly of silicon or germanium in pure or doped forms.
• Conductivity can be manipulated by introducing impurities, a process known as doping.
• Essential in the function of various electronic components such as transistors, diodes, and thermistors.

In the context of a digital thermometer, the thermistor acts as a temperature sensor by providing a measurable change in resistance for a change in temperature, aiding in accurate temperature measurement.

Digital Thermometer

A digital thermometer leverages electrical changes in a thermistor to provide accurate temperature readings. It essentially converts the physical temperature into an electrical signal that can be easily measured and displayed.

Key features of a digital thermometer include:

• Using a thermistor as the temperature-sensing element.
• The capability to provide quick and accurate temperature readings.
• Fan usability often found in medical, industrial, and home environments.

The thermistor inside a digital thermometer is vital as it provides a change in resistance that can be easily monitored. This digital output is then processed to provide a user-friendly temperature reading.

Understanding the interactions of the thermistor's resistance with temperature changes, guided by the temperature coefficient of resistivity, allows for precise readings. Digital thermometers use this relationship to determine body temperature, as seen in the given exercise, ensuring practical application in medical scenarios.