Could a force with these properties exist in a parallel universe?

tl;dr: No, it is not possible to have the force proportional to the product of the square roots of the charges.

In an alternate universe there is a force known as the emotion force although the force has nothing to do with emotions. Anything that interacts through the emotion force has an emotion charge that is either positive or negative. Emotion charge is entirely independent of electric charge.

So far, no problem. There's nothing wrong with having different types of charges which are independent of each other. Indeed, already in our world, we have that.

Like emotion charges attract while opposite charges repel.

I have a feeling that this already should give problems, but I can't pinpoint it, so possibly that's no problem either.

The force between two emotion charges is equal to the square root of the product of their emotion charges divided by the distance between them squared multiplied by a constant.

Here is the problem. While the inverse square of the distance is good (indeed, for a long-range force it's the only reasonable choice), the square root of the emotion charge is not.

Consider the following:

You have two objects of equal emotion charge close to each other, and a third one far enough away that you can in good approximation consider them to be at the same place. Let's call the charge of the far-away particle $q_0$, and the charges of the other two particles $q_1$ and $q_2$

Then you can calculate the force on the far-away particle in two different ways:

1. The force is the sum of the forces due to the attraction from each individual object: $$F = C \frac{\sqrt{q_0}\sqrt{q_1}}{r^2} + C \frac{\sqrt{q_0}\sqrt{q_2}}{r^2} = C\frac{\sqrt{q_0}}{r^2}\left(\sqrt{q_1}+\sqrt{q_2}\right)$$

2. The close-together individual objects are considered a combined object, whose charge is then of course $q_1+q_2$. The force between the far-away object and the combined force is therefore $$F = C\frac{\sqrt{q_0}\sqrt{q_1+q_2}}{r^2} = C\frac{\sqrt{q_0}}{r^2}\left(\sqrt{q_1+q_2}\right)$$

Clearly it is not possible for both equations to be fulfilled at the same time, unless one of the charges is zero.

The force carrier for the emotion charge is massless and so the emotion force has an infinite range. The strength of the emotion force is about 1/100,000.

No problems with this (indeed, a massless force carrier is both necessary and sufficient for an $1/r^2$ force).

posted over 8 years ago

celtschk‭

341 reputation 24 102 42 10

Yes.

We can write out the force as $$\mathbf{F}=-k\frac{\sqrt{e_1e_2}}{r^2}\mathbf{u}_r$$ where $\mathbf{u}_r$ is the unit vector in the direction of $\mathbf{r}$, defined as $$\mathbf{u}_r=\frac{\mathbf{r}}{r}$$ Is this possible? Well, it's an inverse-square law (i.e. $\mathbf{F}\propto r^{-2}\mathbf{u}_r$), and we have two of those in our universe (gravity and the electric force, at least in classical approximations). The only problem is, as Alex mention, the square root. If $\text{sgn}(e_1)=-\text{sgn}(e_2)$, then we have the square root of an imaginary number, and thus a vector of imaginary magnitude, which is not a good thing.

This can be fixed, though. Simply modify the equation to be $$\mathbf{F}=-k\frac{\sqrt{|e_1e_2|}\text{sgn}(e_1)\text{sgn}(e_2)}{r^2}\mathbf{u}_r$$ Here, $\text{sgn}(x)$ is the sign function, defined as $$\text{sgn}(x)=\frac{|x|}{x}=\frac{x}{|x|}$$ This gives us a force that is proportional to the square root of the product of the magnitudes of the charges, as intended, while preserving the property that opposites repel, while like charges attract.

On a different note, it's your universe. In many cases, you can do whatever the heck you want. There are some cases where this doesn't hold, but this isn't one of those exceptions.

posted over 8 years ago

HDE 226868‭

586 reputation 61 463 77 49