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? Temperature coefficient of resistance of nichrome averaged over the temperature range involved is 1.70×10–4 °C–1.
Doubt by Nikunj
Solution :
Temperature Coefficient of Resistanance (α)
α=1.70×10–4 °C–1
α=1.70×10–4 °C–1
Case I
T₀=27°C
T₀=27°C
V=230 V
I₀=3.2 A
R₀=V/I
R₀=230/3.2
R₀=71.875 Ω
R₀=71.875 Ω
R₀=71.88 Ω
Case II
T=?
V=230 V
T=?
V=230 V
I=2.8 A
R=230/2.8
R=82.142 Ω
R=82.14 Ω
R=82.142 Ω
R=82.14 Ω
We know,
R=R₀[1+α(T-T₀)]
R/R₀=1+α(T-T₀)
82.142/71.875=1+α(T-27°)
R=R₀[1+α(T-T₀)]
R/R₀=1+α(T-T₀)
82.142/71.875=1+α(T-27°)
1.1428=1+α(T-27°)
1.1428-1=α(T-27°)
0.1428/α=T-27°
0.1428/[1.70×10–4]=T-27°
[1428/17000]×104=T-27°
[1428/17]×10=T-27°
84×10=T-27
840=T-27
1.1428-1=α(T-27°)
0.1428/α=T-27°
0.1428/[1.70×10–4]=T-27°
[1428/17000]×104=T-27°
[1428/17]×10=T-27°
84×10=T-27
840=T-27
840+27=T
T=867°C
T=867°C
Hence, the steady temperature of the heating element is 867°C.
