# Why is Coulomb's force law a $1/r^2$ dependence, while the Coulomb potential has a $1/r$ potential?

My source of confusion comes from reading the following except from my general chemistry textbook,

"The energy of interaction between a pair of ions can be calculated using Coulomb's law"

$$V = \frac{k Q_1 Q_2}{r}$$

Now, as I learned coulumbs law in general physics, the quantity is (scalar form),

$$F= \frac{kqq}{r^2}$$

How can I rationalize the difference here? Why is potential energy $$1/r$$ dependence, but Coulomb's law a $$1/r^2$$ dependence?

The force (vector) is given by the gradient of the potential energy: $$\vec F = -\vec\nabla V.$$ Even without calculating the gradient explicitly, dimensional analysis (that is, consistency of the units) implies $$|\vec F|\propto \frac{V}{r}$$ with an $$r$$-indepednent coefficient, because the gradient $$\vec \nabla$$ has the same units as $$1/r$$. Therefore, $$V\propto 1/r$$ implies $$|\vec F|\propto 1/r^2$$. This explains the difference. If desired, the details can be filled in using the identity $$\vec\nabla r = \frac{\vec x}{r}$$ with $$r\equiv |\vec x|$$.

• Thanks! That't really helpful. I think the $\vec{F} = - \vec{\nabla} V$ is what I was missing... that makes sense. – jimhacklebarth Jan 27 at 21:49

The voltage is the work done per unit charge so $$V=\frac{1}{q}\int \vec F \cdot d\vec r$$ so if $$F\sim ~1/r^2$$ then $$V\sim \int \frac{dr}{r^2}\sim 1/r$$.

Note that, in the cylindrical geometry - say an infinite line of charge - $$F\sim 1/\rho$$ so $$V\sim \int \frac{d\rho}{\rho}\sim \log(\rho)$$, which brings up interesting when one integrates to $$0$$ or from $$\infty$$.

It is simply because they are different concepts. One is a force and the other is potential energy. Take, for example, a spring where $$F=-kx$$ but $$U=-\frac{1}{2}kx^2$$. They are related, in the one dimensional case by $$-\frac{d U}{d x}=F$$. This becomes the gradient in three dimensions. If this doesn't seem to make sense to you, consider learning mechanics and not looking deeper into electromagnetism.

• I'm not sure how this answers the question, which is specific to the electric case. – ZeroTheHero Jan 28 at 4:22
• Just trying to emphasize that there is no reason to believe a potential energy and force would have the same spatial dependence. Believing in that is indicating a deeper problem in the understanding of mechanics – Aakash Lakshmanan Jan 28 at 4:23
• But this is not a mechanics problem...Sorry but I don't see why discussing springs is relevant to this question on Coulomb's law and potential. Anyways... it's just me. – ZeroTheHero Jan 28 at 4:26
• He points out an energy and asks why it’s different from the force. I’m pointing out that the two, even in the most simple of scenarios, are different and are actually, by definition, different. That is all you need to rationalize they are different. – Aakash Lakshmanan Jan 28 at 4:29