Question

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

Answer 1

Correct Answer: 1027

The resistance of a heating element as a function of temperature is given by \(R = R_0 (1 + \alpha \Delta T)\), where \(R_0\) is the initial resistance, \(\alpha\) is the temperature coefficient, and \(\Delta T\) is the change in temperature. We aim to determine the temperature at which the resistance is \(62 \, \Omega\).

Given values are:

• Initial resistance, \( R_0 = 50 \, \Omega \)
• Final resistance, \( R = 62 \, \Omega \)
• Temperature coefficient, \(\alpha = 2.4 \times 10^{-4} \, ^\circ\text{C}^{-1}\)
• Initial temperature, \( T_0 = 27^\circ\text{C} \)

The change in temperature, \(\Delta T\), is calculated as follows:

\[\begin{align*} 62 &= 50(1 + 2.4 \times 10^{-4} \Delta T) \\ \frac{62}{50} &= 1 + 2.4 \times 10^{-4} \Delta T \\ 1.24 - 1 &= 2.4 \times 10^{-4} \Delta T \\ 0.24 &= 2.4 \times 10^{-4} \Delta T \\ \Delta T &= \frac{0.24}{2.4 \times 10^{-4}} = 1000^\circ\text{C} \end{align*}\]

The final temperature \(T\) is then found by:

\[T = T_0 + \Delta T = 27 + 1000 = 1027^\circ\text{C}\]

The calculated temperature is \(1027^\circ\text{C}\). Therefore, the temperature of the element when its resistance is \(62 \, \Omega\) is \(\boxed{1027^\circ\text{C}}\).