At room temperature (27.0 °C) the resistance of a heating element is 100 Ω. What is the temperature of the element if the resistance is found to be 117 Ω, given that the temperature coefficient of the material of the resistor is \(1.70 \times 10^{-4} °C^{-1}.\)
At room temperature (27.0 °C) the resistance of a heating element is 100 Ω. What is the temperature of the element if the resistance is found to be 117 Ω, given that the temperature coefficient of the material of the resistor is \(1.70 \times 10^{-4} °C^{-1}.\)
Solution and Explanation
Room temperature, T = 27 °C
Resistance of the heating element at T, R = 100 Ω
Let \(T_1\) is the increased temperature of the filament.
Resistance of the heating element at \(T_1, R_1 = 117 Ω \)
Temperature co-efficient of the material of the filament,
\(α = 1.70\times 10^{-4} \degree C^{-1}\)
α is given by the relation,
\(α = \frac{R_1-R}{R(T_1-T)}\)
\(T_1-T =\frac{ R_1-R}{Rα}\)
\(T_1-27 = \frac{117 - 100}{100( 1.70\times10^{-4})}\)
\(T_1-27 = 1000\)
\(T_1 = 1027 °C\)
Therefore, at 1027°C, the resistance of the element is 117Ω.
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Concepts Used:
Temperature Dependence of Resistance
The temperature dependence of resistance is a fundamental property of all materials that conduct electricity. Generally, the resistance of a conductor increases with an increase in temperature. This phenomenon is known as a positive temperature coefficient of resistance.
The reason for this temperature dependence of resistance is related to the interaction of electrons with the crystal lattice of the material. At lower temperatures, the lattice vibrations are minimal, and the electrons are free to move through the material with minimal scattering. This results in a low resistance to the flow of current. However, as the temperature increases, the lattice vibrations increase, causing the electrons to scatter more frequently, which increases resistance.
This phenomenon is governed by the relationship between resistance and temperature known as the temperature coefficient of resistance. The temperature coefficient of resistance is defined as the rate at which resistance changes with respect to temperature. The temperature coefficient of resistance is positive for most metals and semiconductors, meaning that resistance increases with increasing temperature.
However, there are a few materials, such as carbon and certain semiconductors, which exhibit a negative temperature coefficient of resistance. In these materials, the resistance decreases as the temperature increases.
The temperature dependence of resistance has important practical implications in the design and operation of electrical circuits and devices. For example, it is essential to consider the effect of temperature on the resistance of electronic components to ensure reliable and efficient operation of devices over a range of temperatures.