A heating element using nichrome connected to a \(230\;V\) supply draws an initial current of \(3.2\) A which settles after a few seconds to a steady value of \(2.8\;A.\) What is the steady temperature of the heating element if the room temperature is \(27.0 °C\)? Temperature coefficient of resistance of nichrome averaged over the temperature range involved is \(1.70 \times 10-4 °C-1.\)
Answer
Here, the voltage \(V\) given is \(230\;V\). The two currents given are \(I_1\;\&\;I_2\) as \(3.2\;A\) and \(2.8\;A\).
\(V=230\;V\)
\({I_1}=3.2\;A\)
\({I_2}=2.8\;A\)
\({T_1}=27{}^{\circ}C\)
\(\alpha =1.7\times10^{-4}{}^{\circ}C\)
To identify resistance \((R_1)\) before the settlement-
\(V={I_1R_1}\)
\({R_1}=\frac V{{I_1}}\)
\({R_1}=\frac{230}{3.2}\)
\({R_1}=71.88\;\text{ohms}\)
To identify resistance \((R2)\) after the settlement-
\(V={I_2R_2}\)
\({R_2}=\frac V{{I_2}}\)
\({R_2}=\frac{230}{3.2}\)
\({R_2}=82.14\;\text{ohms}\)
We have the relation-
\({R_2}={R_1.}[1+{\alpha .}({T_2}-{T_1})]\)
\(82.14=71.88\{1+1.7\times10^{-4}({T_2}-27)\}\)
\({T_2}=867{}^{\circ}C\)
\(V=230\;V\)
\({I_1}=3.2\;A\)
\({I_2}=2.8\;A\)
\({T_1}=27{}^{\circ}C\)
\(\alpha =1.7\times10^{-4}{}^{\circ}C\)
To identify resistance \((R_1)\) before the settlement-
\(V={I_1R_1}\)
\({R_1}=\frac V{{I_1}}\)
\({R_1}=\frac{230}{3.2}\)
\({R_1}=71.88\;\text{ohms}\)
To identify resistance \((R2)\) after the settlement-
\(V={I_2R_2}\)
\({R_2}=\frac V{{I_2}}\)
\({R_2}=\frac{230}{3.2}\)
\({R_2}=82.14\;\text{ohms}\)
We have the relation-
\({R_2}={R_1.}[1+{\alpha .}({T_2}-{T_1})]\)
\(82.14=71.88\{1+1.7\times10^{-4}({T_2}-27)\}\)
\({T_2}=867{}^{\circ}C\)
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