Some digital thermometers measure the current through a...

Some digital thermometers measure the current through a semiconductor to determine a patient's temperature. If a thermometer uses a germanium wire that has a resistance of $R$ at $37.0^{\circ} \mathrm{C}$ (normal body temperature), what is its resistance at $40.0^{\circ} \mathrm{C} ?$

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Key Concepts

Temperature Coefficient of Resistance

This concept describes how a material’s resistance changes as its temperature changes. It is typically represented by a coefficient that quantifies the percentage change in resistance per unit change in temperature. This coefficient is crucial for understanding and predicting the behavior of materials, especially in temperature-sensitive applications like thermometry.

Resistance-Temperature Relationship

This relationship is captured by equations that relate a material's resistance at a given temperature to its baseline resistance and changes in temperature. It forms the fundamental principle behind using electrical resistance as a means to measure temperature, as small changes in temperature produce measurable changes in resistance.

Semiconductor Sensor Behavior

Semiconductors, such as germanium, exhibit resistance that varies significantly with temperature, often in a nonlinear fashion. In many cases, the resistance of a semiconductor decreases with increasing temperature due to an increase in charge carrier density, which is a key characteristic utilized in designing precise temperature sensors like thermistors.

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