Thermistors, resistors whose resistance is a sensitive function of temperature, are widely used in industry and consumer devices to measure temperature. The resistance of a thermistor at temperature T can be modeled as R=R₀exp[β(1/T−1/T₀)], where T₀ is a reference temperature, the temperatures are in K, and β is a constant with units of K. Suppose you connect a thermistor to a 10.0 V battery and measure the current through it at different temperatures. At 25.0°C, which you select as your reference temperature, the current is 10.0 mA. Raising the temperature to 30.0°C causes the current to increase to 12.5 mA. What is the value of β?

검증된 단계별 안내

Step 1: Start by converting the given temperatures from Celsius to Kelvin. Use the formula T(K) = T(°C) + 273.15. For the reference temperature T₀ = 25.0°C, calculate T₀ in Kelvin. Similarly, calculate T for 30.0°C.

Step 2: Write down the given resistance-temperature relationship for the thermistor: R = R₀exp[β(1/T − 1/T₀)]. Note that the resistance R can be related to the current I and voltage V using Ohm's Law: R = V/I. Use this to express R₀ and R in terms of the given currents and voltage.

Step 3: Calculate R₀ at the reference temperature T₀ using Ohm's Law: R₀ = V/I₀, where I₀ is the current at T₀ (10.0 mA). Similarly, calculate R at T = 30.0°C using R = V/I, where I is the current at 30.0°C (12.5 mA).

Step 4: Substitute the values of R₀, R, T, and T₀ into the resistance-temperature equation: R = R₀exp[β(1/T − 1/T₀)]. Take the natural logarithm of both sides to linearize the equation: ln(R/R₀) = β(1/T − 1/T₀).

Step 5: Solve for β by isolating it in the equation: β = ln(R/R₀) / (1/T − 1/T₀). Substitute the calculated values of R, R₀, T, and T₀ into this equation to determine β.

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주요 개념

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Thermistors

Thermistors are temperature-sensitive resistors that exhibit a significant change in resistance with temperature variations. They are commonly used in temperature measurement and control applications. The resistance of a thermistor decreases with an increase in temperature for NTC (Negative Temperature Coefficient) thermistors, which is the type referenced in the question.

Exponential Resistance Model

The resistance of a thermistor can be modeled using the equation R = R₀exp[β(1/T - 1/T₀)], where R₀ is the resistance at the reference temperature T₀, and β is a material-specific constant. This model illustrates how resistance changes exponentially with temperature, allowing for precise temperature measurements based on current flow through the thermistor.

Ohm's Law

Ohm's Law states that the current (I) flowing through a conductor between two points is directly proportional to the voltage (V) across the two points and inversely proportional to the resistance (R) of the conductor, expressed as V = IR. This principle is essential for analyzing the current through the thermistor when connected to a voltage source, as it allows for the calculation of resistance based on measured current and voltage.

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