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

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,

&\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{aligned}$$ **So, the value of the steady temperature of the nichrome element is $$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.