Let's denote the force between the two particles as F and the distance between them as d. We know that the force is inversely proportional to the square of the distance, which can be expressed as:
F ∝ 1/d^2
Using proportionality, we can write:
F = k/d^2
where k is a constant of proportionality.
If the distance between the particles is 2r, then we have:
F = k/(2r)^2 = k/4r^2
If the distance between the particles is 3r, then we have:
kF = k/d^2 = k/(3r)^2 = k/9r^2
We can then set these two expressions equal to each other and solve for k:
k/4r^2 = k/9r^2
Multiplying both sides by 36r^2 (the least common multiple of 4r^2 and 9r^2), we get:
9k = 4k
Simplifying, we get:
k = 4/9
Therefore, the value of k is 4/9.