# 3.8 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 $1.70\times 10^{-4}\;\degree C^{-1}$.

For the given voltage, the two values of current will correspond to two different values of resistance which will correspond to two different temperature.

Given,

Voltage, V = 230 V

$I_1= 3.2 A$  and $I_2 = 2.8 A$

Using Ohm's law:

$R_1 = 230/3.2 = 71.87\Omega$

and

$R_2= 230/2.8 = 82.14 \Omega$

Now, the temperature coefficient of filament,

$\alpha$ = $1.70\times 10^{-4}\degree C^{-1}$

$\dpi{100} T_1= 27\degree C$

Let $\dpi{100} T_2$ be the steady temperature of the heating element.

We know,

$\dpi{100} R_2 = R_1[1 + \alpha \Delta T]$

$\dpi{80} \implies$ 230/2.8 = 230/3.2[1 + ($\dpi{80} 1.70\times 10^{-4}$) ( T2 -  27) ]

$\dpi{80} \implies$ 3.2/2.8 = 1 + ($\dpi{80} 1.70\times 10^{-4}$) ( T2 -  27)

$\dpi{80} \implies$ 1.14 - 1 =  ($\dpi{80} 1.70\times 10^{-4}$) ( T2 -  27)

$\dpi{80} \implies$ T2 -  27 = 0.14 / ($\dpi{80} 1.70\times 10^{-4}$)

$\dpi{80} \implies$ T2  = (840.5 + 27) °C

Hence, Steady temperature of the element is 867.5 °C.

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