I am looking for confirmation if I've built my equation properly.

My goal is to describe the change in force over time at a given point if evenly distributed radiators (in-phase or cumulative energy/force) in an effectively infinite volume begin emitting simultaneously. One might visualize this as a matrix of light bulbs, but I am strictly calculating for non-interfering radiation (no phase considerations).

Sir Isaac Newton’s inverse square law ($F=\frac{1}{r^2}$) describes the reduction of power from a given source over distance. But I am interested in the influence of multiple sources on a point in the center, so I'm looking at the geometry of progressive spheres with equal surface density of sources ($A=4r^2$).

Attempting to integrate the trend and apply it to gravitational force (the radiative force of interest, 'though any non-interfering radiation should work), my result is the following equation. NOTE: I have not fully simplified it intentionally to show the components used:

$F_{total}=\int_{r_{min}}^{r_{max}}G\frac{M(M4\pi r^2)}{r^2}dr$

I expect the change in the difference in the range of $r$ has a direct linear relationship to $F_{total}$. This should be the case for any set of $r_{min}$ and $r_{max}$ values. The trend I hope to model is, if all radiative sources simultaneously begin emitting, $r_{min}=0$ and $r_{max}=tc$ for a given period of time.

Again, I'm looking for constructive criticism of my approach and any advice on how to correct or further enhance this equation. Thank you.



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