This law-
And the law that energy cannot be created or destroyed.
You can only control the amount of radiation produced by an object by adding or subtracting to the amount of energy available to be converted to radiation.
Any object above absolute zero will attempt to cool down by radiating. It always radiates but it only cools if it emits more energy than it receives.
That law states no such thing...that law states that a perfect black body radiating in a vacuum devoid of any other matter radiates according to its temperature to the 4th power...an ideal, perfect black body in a perfectly empty universe...a fantasy.
Bring matter into the equation and j starts changing in accordance with the difference between the temperature of the radiator and its cooler surroundings. You keep forgetting the basics and jumping to knee jerk conclusions...think ian, I know that you are capable of it.
There is no law that says that the rate of energy emitted by an object can not change if its environment changes...quite the contrary..the application of the SB law relating to radiators that are not perfect black bodies radiating into perfectly empty space says exactly the opposite...P keeps changing as the difference between the temperature of the radiator and the temperature of its surroundings change.
This equation is obviously not for a perfect Blackbody because it has a term for emmisivity.
You should also note that it has no term for area. It is an intensive property.
If you divide the
j by area then you get an actual amount of power, an extensive property. But you then have to do the calculations for every angle that is in line-of-sight of the area you defined.
Your confusion seems to stem from confusing the intensive property of
j with the extensive property of P.
The object exists and radiates. The environment that surrounds the object does not change the radiation, which is always proportional to the temperature of the object.
We can turn the intensive property of radiation into the extensive property of power by defining the radiating surface area of the object. We can turn the intensive property of temperature into the extensive property of total heat content by measuring the mass and the specific heat for its constituents.
With the two extensive properties of power and heat content we can calculate the rate of cooling as time passes. Quickly at first and then progressively slower.
If we add a second object nearby, it also radiates in the same fashion as the first. Some of the radiation from the first will be intercepted by the second, and vice versa. The amount will be determined by the line-of-sight connection. The amount of radiation energy absorbed will add to the total energy content of the two objects, thus slowing the rate of their cooling. But both objects will still be cooling overall.
I will leave it there for now.