A baking dish is removed from a hot oven and placed on a cooling rack. As the dish cools down to 33° C from 181° C, its net radiant power decreases to 13.3 W. What was the net radiant power of the baking dish when it was first removed from the oven? Assume that the temperature in the kitchen remains at 19° C as the dish cools down.
To find the net radiant power of the baking dish when it was first removed from the oven, we can use the Stefan-Boltzmann law.
The Stefan-Boltzmann law states that the net radiant power emitted by an object is proportional to the fourth power of its temperature:
P ∝ T^4
Where P is the net radiant power and T is the temperature.
We can set up a proportion to find the initial net radiant power (P1) when the temperature (T1) was 181°C:
(P1 / P2) = (T1^4 / T2^4)
where P2 is the net radiant power when the temperature is 33°C (13.3 W) and T2 is 33°C.
Let's solve for P1:
(P1 / 13.3) = (181^4 / 33^4)
P1 = 13.3 * (181^4 / 33^4)
P1 ≈ 2470.47 W
Therefore, the net radiant power of the baking dish when it was first removed from the oven was approximately 2470.47 W.
To find the net radiant power of the baking dish when it was first removed from the oven, we can use the Stefan-Boltzmann law, which states that the radiant power emitted by an object is proportional to the fourth power of its temperature.
The Stefan-Boltzmann law is given by:
P = εσA(T^4 - T0^4)
Where:
P = Net radiant power
ε = Emissivity of the object (assumed to be 1 for simplicity)
σ = Stefan-Boltzmann constant (5.67 x 10^-8 W/m^2K^4)
A = Surface area of the object
T = Temperature of the object (in Kelvin)
T0 = Temperature of the surroundings (in Kelvin)
In this case, we need to find the net radiant power (P) when the temperature (T) is 181°C (or 454 K) and the surrounding temperature (T0) is 19°C (or 292 K).
First, let's convert the temperatures to Kelvin by adding 273 to the Celsius temperatures:
T1 = 181°C + 273 = 454 K
T0 = 19°C + 273 = 292 K
Next, we can rearrange the formula to solve for P:
P = εσA(T^4 - T0^4)
Given that the net radiant power when the dish cools down to 33°C (or 306 K) is 13.3 W, we can substitute the values into the equation:
13.3 = 1 * (5.67 x 10^-8) * A * (454^4 - 292^4)
Now, we can solve for A:
A * (454^4 - 292^4) = 13.3 / (5.67 x 10^-8)
A * (454^4 - 292^4) = 2.34642 x 10^11
A = (2.34642 x 10^11) / (454^4 - 292^4)
A ≈ 0.092 m^2 (rounded to three decimal places)
Finally, we can substitute the value of A back into the formula to find the initial net radiant power (P):
P = 1 * (5.67 x 10^-8) * 0.092 * (454^4 - 292^4)
P ≈ 2714.23 W (rounded to two decimal places)
Therefore, the net radiant power of the baking dish when it was first removed from the oven was approximately 2714.23 W.