To determine the temperature of the other patch of skin, we can utilize the Stefan-Boltzmann law, which describes how the power radiated by a black body is proportional to the fourth power of its absolute temperature. Since both patches have an emissivity of 1 (which means they behave like perfect black bodies), we can simplify our calculations. Let's break this down step by step.
Understanding the Problem
We have two patches of skin, both with the same area (A). The patch with the lower temperature is at 300 K, and the power radiated from the two patches differs by 2%. We need to find the temperature of the higher patch.
Applying the Stefan-Boltzmann Law
The Stefan-Boltzmann law states that the power radiated by a black body is given by:
P = εσAT^4
Where:
- P = power radiated
- ε = emissivity (1 for both patches)
- σ = Stefan-Boltzmann constant (approximately 5.67 x 10^-8 W/m²K⁴)
- A = area of the surface
- T = absolute temperature in Kelvin
Calculating the Power Radiated
For the lower temperature patch (T1 = 300 K), the power radiated (P1) can be expressed as:
P1 = σA(300)^4
For the higher temperature patch (T2), the power radiated (P2) is:
P2 = σA(T2)^4
Setting Up the Equation
According to the problem, the difference in power radiated between the two patches is 2%. This can be expressed mathematically as:
P2 = P1 + 0.02P1
Which simplifies to:
P2 = 1.02P1
Substituting the Power Expressions
Now we can substitute the expressions for P1 and P2 into this equation:
σA(T2)^4 = 1.02(σA(300)^4)
Since σ and A are common to both sides, they can be canceled out:
(T2)^4 = 1.02(300)^4
Calculating T2
Next, we need to calculate (300)^4:
(300)^4 = 8.1 x 10^8
Now substituting this value back into the equation:
(T2)^4 = 1.02 x 8.1 x 10^8
(T2)^4 = 8.262 x 10^8
To find T2, we take the fourth root of both sides:
T2 = (8.262 x 10^8)^(1/4)
Calculating this gives:
T2 ≈ 301.5 K
Final Result
The temperature of the other patch of skin is approximately 301.5 K. This small increase in temperature reflects the 2% difference in the rates of heat radiation between the two patches, illustrating how even minor changes in temperature can significantly affect radiative heat transfer.
