An electric toaster uses nichrome for its heating element. When a… - Vidyadip

Question

An electric toaster uses nichrome for its heating element. When a negligibly small current passes through it, its resistance at room temperature $\left(27.0^{\circ} C \right)$ is found to be $75.3 \Omega$. 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 $1.70 \times 10^{-4}{ }^{\circ} C ^{-1}$.

✓

Answer

When the current through the element is very small, heating effects can be ignored and the temperature $T_1$ of the element is the same as room temperature. When the toaster is connected to the supply, its initial current will be slightly higher than its steady value of $2.68 A$. But due to heating effect of the current, the temperature will rise. This will cause an increase in resistance and a slight decrease in current. In a few seconds, a steady state will be reached when temperature will rise no further, and both the resistance of the element and the current drawn will achieve steady values. The resistance $R_2$ at the steady temperature $T_2$ is
$
R_2=\frac{230 V }{2.68 A }=85.8 \Omega
$
Using the relation
$
R_2=R_1\left[1+\alpha\left(T_2-T_1\right)\right]
$
with $\alpha=1.70 \times 10^{-4}{ }^{\circ} C ^{-1}$, we get
$
T_2-T_1=\frac{(85.8-75.3)}{(75.3) \times 1.70 \times 10^{-4}}=820^{\circ} C
$
that is, $T_2=(820+27.0){ }^{\circ} C =847{ }^{\circ} C$
Thus, the steady temperature of the heating element (when heating effect due to the current equals heat loss to the surroundings) is $847^{\circ} C$.

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