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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 27.0°C? The temperature coefficient of resistance of nichrome averaged over the temperature range involved is 1.70 × 10−4 °C−1.
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Solution
Given: Supply voltage, V = 230 V
Initial current drawn, I1 = 3.2 A
Initial resistance = R1
`R_1 = V/I`
= `230/3.2`
= 71.87 Ω
Steady-state current, I2 = 2.8 A
Resistance at the steady state = R2
R2 = `230/2.8`
= 82.14 Ω
Temperature coefficient of nichrome,
α = 1.70 × 10−4 °C−1
Initial temperature of nichrome,
T1 = 27.0°C
Let the steady-state temperature be T2.
T2 can be obtained by the relation for α,
α = `(R_2 - R_1)/(R_1(T_2 - T_1))`
`1.7 xx 10^-4 = (82.14 - 71.87)/(71.87(T_2 - 27°C))`
`T_2 − 27°C = (82.14 - 71.87)/(71.87 xx 1.7 xx 10^-4)`
`T_2 − 27°C = 10.27/(122.18 xx 10^-4)`
`T_2 − 27°C = 10.27/0.012218`
T2 − 27°C = 840.5
T2 = 840.5 + 27
T2 = 867.5°C
Therefore, the steady temperature of the heating element is 867.5°C.
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