Question

A heating element using nichrome connected to a $230V$ supply draws an intial current of $3.2A$ which settles after a few seconds to a steady value of $2.8A.$ 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 tempature range involved is $1.70\times {10}^{-4}$ ${}^{\circ }{C}^{-1}$

Answer 1

Step1: Find intial resistance.

Given, supply voltage $V=230V$
Initial temperature $T_1={27}^{\circ }C$
Intial current drawn at temperature $T_1,I_1=3.2A$

Intial resistance at temperature $T_1,R_1=\frac{V}{I_1}$

$R_1=\frac{230}{3.2}=71.87\Omega$

Step 2 : find resistance at steady state temperature.

Let steady state temperature be $T_2$

Steady state value of current at temperature $T_2,I_2=2.8A$

Resistance at temperature $T_2,R_2=\frac{V}{I_2}$

$R_2=\frac{230}{2.8}=82.14\Omega$

$Step3:$ Find steady state temperature.

Given, temperature coefficient of nichrome, $\alpha =1.70\times {10}_{-6}$ ${}^{\circ }{C}^{-1}$

Using the formula $R_2=R_1(1+\alpha (T_2-T_1\left)\right)$

$T_2=T_1+\frac{R_2-R_1}{\alpha R_1}$

$T_2=27+\frac{82.14-71.87}{1.70\times {10}^{-4}\times 71.87}$

$T_2=27+840.5$

$T_2=867.5$ ${}^{\circ }C$

Final answer: Steady temperature of the

heating element is $867.5$${}^{\circ }C.$