How much does the luminosity of a spherical source of blackbody radiation increase if you triple its size?
Luminosity is proportional to area. I am not certain what you mean by "size", if that is area, or diameter.
To calculate the change in luminosity when you triple the size of a spherical source of blackbody radiation, you can use the formula for the luminosity of a blackbody:
L = 4πR²σT⁴,
where L is the luminosity, R is the radius of the source, σ is the Stefan-Boltzmann constant, and T is the temperature.
To find the change in luminosity, we need to compare the luminosity of the original source (L₁) to the luminosity of the new, larger source (L₂).
If you triple the size of the spherical blackbody source, the new radius (R₂) will be three times the original radius (R₁). So, R₂ = 3R₁.
Using the formula, we have:
L₁ = 4πR₁²σT⁴,
L₂ = 4πR₂²σT⁴.
Substituting the value of R₂:
L₂ = 4π(3R₁)²σT⁴,
L₂ = 4π9R₁²σT⁴.
We can see that the luminosity of the new, larger source (L₂) is proportional to the square of the original radius (R₁). So, if you triple the size of the source, the luminosity will increase by a factor of 9.
In other words, the luminosity will increase by 900% (9 times) when you triple the size of a spherical source of blackbody radiation.