A heating element using nichrome connected to a 230 V supply draw… - Vidyadip

Question

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${ }^{\circ} C$ ? Temperature coefficient of resistance of nichrome averaged over the temperature range involved is $1.70 \times 10^{-4}{ }^{\circ} C ^{-1}$.

✓

Answer

Given that :
Magnitude of supply voltage, V $=230 V$
Initial current, $I _1=3.2 A$
Room temperature $T _1=27^{\circ} C$
Constant current, $I _2=2.8 A$
Constant temperature $T _2=$ ?
Temperature coefficient of resistance
$\alpha=1.7 \times 10^{-4}{ }^{\circ} C ^{-1}$
If $R_1$ and $R_2$ are resistances of the wire at temperatures $T_1$ and $T_2$ respectively, then
Formula $\quad R _1=\frac{ V }{ I _1}=\frac{230}{3.2}=71.875 \Omega$
$R _2=\frac{ V }{ I _2}=\frac{230}{2.8}=82.143 \Omega$
Formula $\quad \alpha=\frac{ R _2- R _1}{ R _1\left(T_2- T _1\right)}$
$T _2- T _1=\frac{ R _2- R _1}{ R _1 \alpha}$
$\Rightarrow \quad T _2-27^{\circ} C =\frac{82.143-71.875}{71.875 \times 1.7 \times 10^{-4}}=\frac{10.268}{122.19 \times 10^{-4}}$
$\Rightarrow \quad T _2-27^{\circ} C =\frac{10.268}{122.19 \times 10^{-4}}=840.35$
$T _2=840.35+27=867.35^{\circ} C$

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