Three spheres are made from the same material, but each has a different size and temperature. The radii of spheres A, B, and C are 1.5R, R, and 1.25R, respectively. Which of the following choices gives the correct ranking of the spheres in order of least radiant energy emitted per second to the most radiant energy emitted per second?
Entry field with incorrect answer
B A C
C A B
C B A
A C B
B C A
To determine the ranking of the spheres in terms of the radiant energy emitted per second, we need to consider two factors: the surface area of each sphere and the Stefan-Boltzmann law, which relates the radiant energy emitted by an object to its temperature.
The surface area of a sphere is given by the formula A = 4πr^2, where r is the radius of the sphere.
Now, let's examine the spheres A, B, and C:
Sphere A has a radius of 1.5R, so its surface area is A_A = 4π(1.5R)^2 = 9πR^2.
Sphere B has a radius of R, so its surface area is A_B = 4πR^2.
Sphere C has a radius of 1.25R, so its surface area is A_C = 4π(1.25R)^2 = 5πR^2.
According to the Stefan-Boltzmann law, the radiant energy emitted per second (E) is proportional to the surface area (A) of the object multiplied by the fourth power of its temperature (T^4).
Now, let's compare the temperatures:
Assuming the temperatures of the spheres are T_A, T_B, and T_C, and considering that they are at different temperatures, we can simply compare their temperatures rather than calculating their actual values.
Given that:
T_A > T_B > T_C
According to the Stefan-Boltzmann law, the radiant energy emitted per second will be:
E_A ∝ A_A * T_A^4
E_B ∝ A_B * T_B^4
E_C ∝ A_C * T_C^4
Since E ∝ A * T^4, we can see that the radiant energy emitted per second is proportional to both the surface area and the temperature raised to the fourth power.
Now, let's compare the values of A and T for spheres A, B, and C:
A_A > A_B > A_C
T_A^4 > T_B^4 > T_C^4
From these comparisons, we can conclude that sphere A will emit the most radiant energy per second, followed by sphere B, and then sphere C.
Therefore, the correct ranking of the spheres in order of least radiant energy emitted per second to the most radiant energy emitted per second is:
C > B > A
Hence, the correct answer is C B A.