A certain lightbulb has a tungsten filament with a resistance of 28 Ω when cold and 144 Ω when hot.
If the equation R = R0 [1 + α ∆T ]
can be used over the large temperature range involved here, find the temperature of the fila- ment when it is hot. Assume that α , the tem- perature coefficient of resistivity of tungsten, is0.0045(◦C)−1 andthatthetemperatureof the cold filament is 17◦C.
Answer in units of ◦C.
To find the temperature of the filament when it is hot, we can use the given equation:
R = R0 [1 + α ∆T]
Where:
R = Resistance when hot = 144 Ω
R0 = Resistance when cold = 28 Ω
α = Temperature coefficient of resistivity of tungsten = 0.0045 (◦C)⁻¹
∆T = Change in temperature
We are given the resistance values, and we need to find the corresponding change in temperature (∆T). Let's rearrange the equation to solve for ∆T:
R = R0 [1 + α ∆T]
R / R0 = 1 + α ∆T
Let's substitute the values we have:
144 Ω / 28 Ω = 1 + 0.0045 (◦C)⁻¹ ∆T
5.1429 = 1 + 0.0045 (∆T)
Now, let's isolate ∆T by subtracting 1 from both sides:
5.1429 - 1 = 0.0045 (∆T)
4.1429 = 0.0045 (∆T)
Finally, to find ∆T, we divide both sides of the equation by 0.0045:
∆T = 4.1429 / 0.0045
∆T ≈ 920.64 (◦C)
Now, to find the temperature of the filament when it is hot, we add ∆T to the initial temperature of the cold filament (17 ◦C):
Temperature when hot = 17 ◦C + 920.64 ◦C
Temperature when hot ≈ 937.64 ◦C
Therefore, the temperature of the filament when it is hot is approximately 937.64 ◦C.