Question

An electric toaster uses nichrome for its heating element. When a negligibly small current passes through it, its resistance at room temperature (27.0 °C) is found to be 75.3 Ω. When the toaster is connected to a 230 V supply, the current settles, after a few seconds, to a steady value of 2.68 A. What is the steady temperature of the nichrome element? The temperature coefficient of resistance of nichrome averaged over the temperature range involved, is \[1.7\times {{10}^{-4}}/{}^\circ C\].

Answer 1

Hint:We are given with nichrome wire; its resistance is given at one temperature and also value at a given voltage. We are also given a temperature coefficient of resistance. So, we can use Ohm’s law and the formula of resistance variation with temperature to arrive at our solution.

Complete step by step answer:
Ohm’s law says, \[V=IR\].
So, at some particular temperature, the resistance variation is given as \[{{R}_{t}}={{R}_{0}}[1+\alpha (T-{{T}_{0}})]\].
\[{{R}_{t}}\] is the temperature at a given temperature and \[{{R}_{0}}\] is the resistance at room temperature.
\[\alpha \]is the temperature coefficient of resistance and T is the given temperature and \[{{T}_{0}}\]is the room temperature.
Using ohm's law, \[R=\dfrac{V}{I}=\dfrac{230}{2.68}=85.82\Omega \].
Now using the second relationship,
\[\begin{align}
&\Rightarrow {{R}_{T}}={{R}_{0}}[1+\alpha (T-{{T}_{0}})] \\
&\Rightarrow 85.82=75.3[1+1.7\times {{10}^{-4}}(T-27)] \\
&\Rightarrow \dfrac{85.82}{75.3}=1+1.7\times {{10}^{-4}}(T-27) \\
&\Rightarrow 1.13=1+1.7\times {{10}^{-4}}(T-27) \\
&\Rightarrow 0.13=1.7\times {{10}^{-4}}(T-27) \\
&\Rightarrow \dfrac{0.13}{1.7\times {{10}^{-4}}}=(T-27) \\
&\therefore T=848.81{}^\circ C \\
\end{align}\]

So, the value of the steady temperature of the nichrome element is \[848.81{}^\circ C\].

Note: We have to be careful while doing the calculations and need to rake all decimal values and not just up to one place. . A positive value of the temperature of coefficient resistance of a material means that its resistance increases with an increase in temperature.Most conductive materials change specific resistance with temperature changes. That is why we have temperature variation of resistance and resistivity.