Question 389: Current Electricity

Question

At room temperature $$(27^{\circ} \mathrm{C})$$, the resistance of a heating element is $$50 \Omega$$. The temperature coefficient of the material is $$2.4 	imes 10^{-4}{ }^{\circ} \mathrm{C}^{-1}$$. The temperature of the element, when its resistance is $$62 \Omega$$, is __________$${ }^{\circ} \mathrm{C}$$.

Accepted Answer

We can start solving this problem by first understanding that the resistance of a material changes with temperature, and this change can be quantified using the temperature coefficient of resistance $ \alpha $. The relationship between the resistance of a material at any temperature $ T $ and its resistance at a reference temperature $ T_0 $ is given by the formula:
$ R = R_0(1 + \alpha(T - T_0)) $
Where:
$ R $ is the resistance at temperature $ T $ (in this case, $ 62 \Omega $).
$ R_0 $ is the resistance at reference temperature $ T_0 $ (in this case, $ 50 \Omega $).
$ \alpha $ is the temperature coefficient of resistance (in this case, $ 2.4 	imes 10^{-4} \, ^\circ\mathrm{C}^{-1} $).
$ T $ is the unknown temperature we need to find.
$ T_0 $ is the reference temperature, given as $ 27^\circ \mathrm{C} $.
By substituting the given values into the formula, we get:
$ 62 = 50(1 + 2.4 	imes 10^{-4}(T - 27)) $
First, divide both sides of the equation by 50:
$ \frac{62}{50} = 1 + 2.4 	imes 10^{-4}(T - 27) $
Then solve for $ T $:
$ 1.24 = 1 + 2.4 	imes 10^{-4}(T - 27) $
$ 0.24 = 2.4 	imes 10^{-4}(T - 27) $
$ \frac{0.24}{2.4 	imes 10^{-4}} = T - 27 $
$ 1000 = T - 27 $
$ T = 1027 ^\circ\mathrm{C} $
Thus, the temperature of the element when its resistance is $ 62 \Omega $ is $ 1027^\circ\mathrm{C} $.

Current Electricity

At room temperature $(27^{\circ} \mathrm{C}), the resistance of a heating element is 50 \Omega. The temperature coefficient of the material is 2.4 \times 10^{-4}{ }^{\circ} \mathrm{C}^{-1}. The temperature of the element, when its resistance is 62 \Omega, is __________{ }^{\circ} \mathrm{C}$.

Choose the correct answer: