# If electromagnetic waves didn't follow an inverse square law?

I was thinking of a hypothetical universe in which electromagnetic waves decrease in intensity with the distance instead of the square of the distance. What kinds of effects would that have on that universe and how would electromagnetism need to be different for this to be the case?

• Could you explain more why you wish to make this change? You can't really change one thing about physics, and hope the rest stays the same - the decrease in intensity is a corollary to the fact that the surface area of a sphere increases with the square of radius. Changing this means we don't get conservation of energy - so we can, for instance, have perpetual motion by putting a light bulb in a sphere of solar cells. But perpetual motion is really bad news for the universe, since it means energy will increase without bound, so everything will probably blow up or burn up. Nov 3 '15 at 4:18
• @MiloBrandt Or it could be a 2D universe, which probably isn't what OP is looking for. Nov 3 '15 at 4:55
• @Milo Brandt One way to get around the problem of conservation of energy for this hypothetical universe would be for the mass and charge of each charged particle to fill all of space but for the mass and charge density of that charged particle to decrease with the square of the distance from its center. Nov 3 '15 at 5:14
• Do you mean that the energy of light gets weaker over time(the frequency stretched a.k.a redshifted)? You need to lose some of its energy or deposits a portion of it momentum somewhere but photon can't decay so tough luck. Nov 3 '15 at 9:53
• As a distillation of the other comments, the inverse square law for EM radiation is due to the dimensionality of the Universe and the "Law" of Conservation of Energy. If the intensity of light followed a $\frac{1}{r}$ ratio, then the total amount of energy in the wave would increase by $r \times$. The only way to avoid increasing energy would be if the Universe were 2D rather than 3D. The Conservation laws ultimate stem from Symmetry which is a fundamental theory of how the Universe works. Nov 3 '15 at 15:22

As a distillation of the comments and simplification of the previous answer, the flux of EM energy decreases according to inverse square law due to the dimensionality of the Universe and the "Law" of Conservation of Energy which stems from the physical theory of Symmetry.

Units for Energy = $J\;_,$

Units for Fluence = $\mathrm{\dfrac{J}{cm^2}}\;_,$

Units for Area = $\mathrm{cm}^2\;_.$

Fluence is defined as Energy divided by Area: $$F = \frac{E}{A} \rightarrow f\left(\frac{E}{r^2}\right) \rightarrow \text{Units: } \mathrm{\frac{J}{cm^2}} = \mathrm{J \div cm^2}$$

Similarly: $$E = F \times A \rightarrow \text{Units: } J = \mathrm{\frac{J}{cm^2} \times cm^2}$$

Meaning the EM Fluence passing through any unit area of the sphere decreases as the inverse square of the radius. Assuming the total energy remains constant (assume the Law of Conservation of Energy is true), this is the origin of the inverse square law.

If you "break" this by changing the equation to inversely proportional, not only does the total amount of energy increase as the EM radiation expands, the units also do not work out. It "breaks" the Universe.

$$E = F \times r \rightarrow \text{Units: } J \not = \mathrm{\frac{J}{cm^2} \times cm = \frac{J}{cm}}$$

If the intensity of light followed a $\frac{1}{r}$ ratio, then the total amount of energy in the wave would increase by $E \times r$. This means the entire 3D universe would be cooked as total energy increased linearly as it propagated further and further from the source.

The only way to avoid increasing energy would be if the Universe were 2D rather than 3D.

If you want to use this relationship as part of your Universe, then you must either move to a 2 dimensional Universe or abandon the Conservation of Energy.

• +1, excellent description. I'm curious why you used centimeters instead of meters though. It's not a big deal, obviously, just not SI. Nov 3 '15 at 16:41
• Habit. Any distance unit works for the description ($furlongs^2$ anyone?) Nov 3 '15 at 16:54

A universe with this law does not support particle physics as we know it. From Wikipedia:

The electromagnetic force is the one responsible for practically all the phenomena one encounters in daily life above the nuclear scale, with the exception of gravity.

For us:

$$\mbox{Intensity} \ \propto \ \frac{1}{\mbox{distance}^2} \,$$

For them:

$$\mbox{Intensity} \ \propto \ \frac{1}{\mbox{distance}} \,$$

Coulomb's Law states that:

The magnitude of the electrostatic force of interaction between two point charges is directly proportional to the scalar multiplication of the magnitudes of charges and inversely proportional to the square of the distance between them.[12]

The force is along the straight line joining them. If the two charges have the same sign, the electrostatic force between them is repulsive; if they have different signs, the force between them is attractive.

The scalar equation describing this law is:

$$\mathbf F|=k_e{|q_1q_2|\over r^2}\qquad$$

In this alternate universe, Coulumb's law looks like this:

$$\mathbf F|=k_e{|q_1q_2|\over r}\qquad$$

The effect of this change is that electrons will be much more repulsed from each other and oppositely charge atoms will attract each other over a much larger distance.

I don't think the changes to the electromagnetic force will make neutronium soup but all the interactions above the nuclear level are going to be really weird. The sun will appear to burn hotter, magnets will stick to each other or repel each other at much greater distances. Friction is going to be much harder to overcome....fluids will be more viscous. The implications to this kind of change are very far reaching. It's just going to be really weird.