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 $27.0^\circ C?$ Temperature coefficient of resistance of nichrome averaged over the temperature range involved is $1.70\times10^{-4}{^\circ\text{C}}^{-1}.$
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Potential applied across the heating element, $V = 230 V$
Initial current,$ l_1 = 3.2 A$
After few seconds, steady current attained is, $l_2 = 2.8 A$
Using ohm's law,
$\text{R}_1=\frac{230}{3.2}=71.87\ \Omega$
and,
$\text{R}_2=\frac{230}{2.8}=82.14\ \Omega$
$\alpha=1.7\times10^{-4}{^\circ\text{C}^{-1}}$
$\text{t}_1=27^\circ\text{C}$
Using the relation,
$\text{R}_2=\text{R}_1[1+\alpha(\text{t}_2-\text{t}_1)]$
$\text{Therefore,}\ \ \text{t}_2=\frac{\text{R}_2-\text{R}_1}{\text{R}_1.\alpha}+\text{t}_1$
$\Rightarrow\ \ \text{t}_2=\frac{82.14-71.87}{71.87\times1.7\times10^{-4}}+27$
$=840.56+27$
$=867.56^\circ\text{C}$
$=867^\circ\text{C}$
Potential applied across the heating element, $V = 230 V$
Initial current,$ l_1 = 3.2 A$
After few seconds, steady current attained is, $l_2 = 2.8 A$
Using ohm's law,
$\text{R}_1=\frac{230}{3.2}=71.87\ \Omega$
and,
$\text{R}_2=\frac{230}{2.8}=82.14\ \Omega$
$\alpha=1.7\times10^{-4}{^\circ\text{C}^{-1}}$
$\text{t}_1=27^\circ\text{C}$
Using the relation,
$\text{R}_2=\text{R}_1[1+\alpha(\text{t}_2-\text{t}_1)]$
$\text{Therefore,}\ \ \text{t}_2=\frac{\text{R}_2-\text{R}_1}{\text{R}_1.\alpha}+\text{t}_1$
$\Rightarrow\ \ \text{t}_2=\frac{82.14-71.87}{71.87\times1.7\times10^{-4}}+27$
$=840.56+27$
$=867.56^\circ\text{C}$
$=867^\circ\text{C}$
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