A body's temperature is increased from 900K to 1900K. By what factor does the total power radiated per unit area increase?
To find the factor by which the total power radiated per unit area increases, we can use the Stefan-Boltzmann law, which states that the total power radiated per unit area by a black body is directly proportional to the fourth power of its absolute temperature.
The formula for the power radiated per unit area (P/A) is given by:
P/A = σ * T^4
Where:
P/A = Power radiated per unit area
σ = Stefan-Boltzmann constant (approximately 5.67 x 10^-8 W/m^2K^4)
T = Absolute temperature
Let's calculate the initial power radiated per unit area (P1/A1) and the final power radiated per unit area (P2/A2) and find the factor by which it increases.
Given:
Initial temperature (T1) = 900 K
Final temperature (T2) = 1900 K
Using the Stefan-Boltzmann law formula, we can calculate both initial and final power radiated per unit area:
P1/A1 = σ * T1^4
P2/A2 = σ * T2^4
Calculating P1/A1:
P1/A1 = σ * T1^4
= 5.67 x 10^-8 W/m^2K^4 * (900K)^4
≈ 2.8657 x 10^4 W/m^2
Calculating P2/A2:
P2/A2 = σ * T2^4
= 5.67 x 10^-8 W/m^2K^4 * (1900K)^4
≈ 1.5151 x 10^6 W/m^2
Now, to find the factor by which the power radiated per unit area increases, we can divide P2/A2 by P1/A1:
Factor = (P2/A2) / (P1/A1)
= (1.5151 x 10^6 W/m^2) / (2.8657 x 10^4 W/m^2)
≈ 52.878
Therefore, the total power radiated per unit area increases by a factor of approximately 52.878 when the temperature is increased from 900K to 1900K.