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Inverse Square Law

The force due to a point charge decreases proportionally to the square of the distance from the charge, a key component of Coulomb's Law.
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The statement of the theorem

Consider two point charges, q1q_1 and q2q_2, located at positions r⃗1\vec{r}_1 and r⃗2\vec{r}_2 in R3\mathbb{R}^3. Let the separation vector be r⃗=r⃗2−r⃗1\vec{r} = \vec{r}_2 - \vec{r}_1, and the distance be r=∣r⃗∣r = |\vec{r}|. The electrostatic force F⃗21\vec{F}_{21} exerted on q2q_2 by q1q_1 is rigorously defined by Coulomb's Law:\n\nF⃗21=kq1q2r2r^=kq1q2∣r⃗∣2r⃗∣r⃗∣\vec{F}_{21} = k \frac{q_1 q_2}{r^2} \hat{r} = k \frac{q_1 q_2}{|\vec{r}|^2} \frac{\vec{r}}{|\vec{r}|} \n\nwhere k=14πϵ0k = \frac{1}{4\pi\epsilon_0} is Coulomb's constant, and r^=r⃗r\hat{r} = \frac{\vec{r}}{r} is the unit vector pointing from q1q_1 to q2q_2. The scalar magnitude of this force, F=∣F⃗21∣F = |\vec{F}_{21}|, explicitly demonstrates the Inverse Square Law:\n\nF=k∣q1q2∣r2=Cr2F = k \frac{|q_1 q_2|}{r^2} = \frac{C}{r^2} \n\nwhere C=k∣q1q2∣C = k |q_1 q_2| is a constant determined by the product of the charges and the permittivity of free space.