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A heating element using nichrome connected to a 230 $V$ supply draws an initial current of $3.2 \mathrm{~A}$ which settles after a few seconds to a steady value of $2.8 \mathrm{~A}$. What is the steady temperature of the heating element if the room temperature is $27.0^{\circ} \mathrm{C}$ ? Temperature coefficient of resistance of nichrome averaged over the temperature range involved is $1.70 \times 10^{-4}{ }^{\circ} \mathrm{C}^{-1}$.
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Verified Answer
Given, $\mathrm{R}_1=\frac{230}{3.2} \Omega$,
$$
\begin{aligned}
&\mathrm{R}_2=\frac{230}{2.8} \Omega, \alpha=1.7 \times 10^{-4{ }^{\circ} \mathrm{C}^{-1}}, \\
&\mathrm{t}_1=27^{\circ} \mathrm{C}, \mathrm{t}_2=?
\end{aligned}
$$
By formula, $\mathrm{R}_2=\mathrm{R}_1\left[\left(1+\alpha\left(\mathrm{t}_2-\mathrm{t}_1\right)\right]\right.$
$$
t_2=\frac{R_2-R_1}{R_1 \alpha}+t_1=\frac{\frac{230}{2.8}-\frac{230}{3.2}}{\frac{230}{3.2} \times 1.7 \times 10^{-4}}+27
$$
$$
=\frac{3.2-2.8}{3.2 \times 2.8 \times \frac{1}{3.2} \times 1.7 \times 10^{-4}}+27
$$
$$
=\frac{0.4}{2.8 \times 1.7 \times 10^{-4}}+27=840.34+27
$$
$$
=867.34^{\circ} \mathrm{C} \text {. }
$$
$$
\begin{aligned}
&\mathrm{R}_2=\frac{230}{2.8} \Omega, \alpha=1.7 \times 10^{-4{ }^{\circ} \mathrm{C}^{-1}}, \\
&\mathrm{t}_1=27^{\circ} \mathrm{C}, \mathrm{t}_2=?
\end{aligned}
$$
By formula, $\mathrm{R}_2=\mathrm{R}_1\left[\left(1+\alpha\left(\mathrm{t}_2-\mathrm{t}_1\right)\right]\right.$
$$
t_2=\frac{R_2-R_1}{R_1 \alpha}+t_1=\frac{\frac{230}{2.8}-\frac{230}{3.2}}{\frac{230}{3.2} \times 1.7 \times 10^{-4}}+27
$$
$$
=\frac{3.2-2.8}{3.2 \times 2.8 \times \frac{1}{3.2} \times 1.7 \times 10^{-4}}+27
$$
$$
=\frac{0.4}{2.8 \times 1.7 \times 10^{-4}}+27=840.34+27
$$
$$
=867.34^{\circ} \mathrm{C} \text {. }
$$
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