Question

At room temperature(27℃) , the resistance of a heating element is 100Ω. What is the temperature of the element if the resistance is found to be 117Ω, given that the temperature coefficient of the material of the resistor is $1.70 \times {10^{ - 4}}{C^{ - 1}}$

Answer 1

Hint: For these type of questions, the formula for resistance values for conductors at any temperature other than the standard temperature (usually specified at room temperature) on the specific resistance table is given by-
$R = {R_0}[1 + \alpha (T - {T_0})]$
R= Conductor resistance at temperature T
${R_0}$=Conductor resistance at reference temperature ${T_0}$
α= Temperature coefficient of resistance for conductor material.
T= Conductor temperature in degree Celsius
${T_0}$ =Reference temperature that α is specified at for the given conducting material.
This is a straightforward application of the formula. In the given problem, we need to find out the temperature of element (T) at a resistance value 117Ω.

Complete step-by-step solution:
Given, Room temperature, ${T_0}$= 27°C
Resistance of the heating element at reference temperature ${T_0}$, ${R_0}$= 100 Ω
Let T is the increased temperature of the element.
Resistance of the heating element at T, R = 117Ω
Putting all the values in the formula to find T, we get
⇒$117 = 100[1 + 1.70 \times {10^{ - 4}}(T - 27)]$
⇒$\dfrac{{117}}{{100}} - 1 = 1.70 \times {10^{ - 4}}(T - 27)$
⇒$\dfrac{{117 - 100}}{{100(1.70 \times {{10}^{ - 4}})}} = T - 27$
⇒$1000 = T - 27$
⇒$T = 1027$℃

Therefore, $T = 1027$℃ is the required temperature of the element.

Note: The “alpha” (α) constant used in formula is known as the temperature coefficient of resistance and symbolizes the resistance change factor per degree of temperature change. They also change resistance of conductor material according to temperature by specific amounts.
For pure metals, this coefficient is a positive number, which means that resistance increases with increasing temperature.
For the elements carbon, silicon, and germanium, this coefficient is a negative number, which means that resistance decreases with increasing temperature.
For some metal alloys, the temperature coefficient of resistance is very close to zero, which means that the resistance hardly changes at all with variations in temperature.