Imagine an infinite space of infinite electric charge.
This scenario leads to a paradox with profound implications.
In order to understand this paradox, we need to understand electric flux.
Electric flux is generated by positively charged particles and absorbed by negatively charged particles.
The density of electric flux is largest near the charged particles,
indicating that this is where the electric field is strongest.
The amount of flux indicates the strength of the electric field.
The direction of the flux indicates the direction of the electric field.
This is only an artistic representation, since the electric flux is continuous throughout space,
and the electric field is not actually flowing.
In fact, electric flux does not even exist,
but is just a mathematical concept to help us calculate the behavior of charged particles.
Particles with the same charge repel one another.
Particles with opposite charges attract one another.
Particles that are nearby have a greater effect than particles further away,
which allows the charged particles in this example to join together.
The density of electric flux is largest near the charged particles,
indicating that this is where the electric field is strongest.
The electric flux passing through a surface depends on the strength of the electric field …
The area of the surface …
And the angle that the surface makes with the electric field.
The surface can also be curved.
The net amount of electric charge inside a closed surface determines the net amount of electric flux entering or exiting.
This remains true regardless of the size or shape of the closed surface.
Particles with opposite charges cancel each other out,
hence any closed surface containing just these two particles
will have an electric flux exiting that is exactly equal to the amount of electric flux entering.
Now consider a scenario where we have a uniform positive charge density evenly spread out through all of space.
Every closed surface throughout this space has a net positive charge inside.
Therefore, every closed surface must have a net amount of electric flux exiting.
But this leads to a number of logical contradictions.
As the volume of our closed surface grows larger, the amount of positive charge inside also grows larger.
The volume grows faster than the surface area.
The math therefore tells us that as we continue to increase the distance away from the center to infinity,
the strength of the electric field also continues to increase to infinity.
But, we can select any other point in this infinite space as the center,
and create a similar mathematical solution around this new point.
We therefore have an infinite number of equally valid mathematical solutions to this scenario, all of which contradict each other.
Though, if we think about it, this should not surprise us.
Suppose we have an infinite space with no charged particles at all.
Having a uniform electric field of constant strength everywhere throughout this infinite space
is a perfectly valid mathematical solution, consistent with the laws of physics.
For any closed surface we create, the electric flux entering will be exactly equal to the electric flux exiting,
correctly indicating that the net electric charge inside the closed surface is zero.
There are an infinite number of mathematical solutions for the electric field,
unless we are given a boundary condition, such as knowing the electric field at one point.
As an analogy, Newton’s Laws of motion will give an infinite number of possible trajectories,
unless we are given an initial position and initial velocity.
In the cases of trying to find the electric field for a limited number of particles,
the boundary condition typically assumed is that the electric field at a point infinitely far away is zero.
In our scenario of the infinite space of infinite charge,
if we specify the electric field at any one point in space,
then we get a unique solution for the electric field at every point in space.
But there is still a paradox.
If we have a continuous charge density evenly spread out through all space,
our intuition tells us that the net electric field should be zero,
since the effects of all the positive charges would cancel each other out, due to the symmetry of space.
But this does not match any of the mathematical solutions,
each of which states that the electric field continues to increase to infinity as we move infinitely far away.
And this is assuming that we actually have an infinite space.
If we have a Closed Universe, where space-time completely wraps around on itself,
then the scenario of a uniform positive charge density has no mathematical solution at all.
For a Closed Universe, the only way we could have a logically consistent solution
is if the total amount of positive charge in the Universe
is exactly equal to the total amount of negative charge in the Universe.
Even a single extra positive or negative particle would make any solution impossible.
On the other hand, for an Infinite Universe,
we could have an infinite amount of extra positive or negative particles, and this would be OK.
As one example, we can have an infinite line of charge.
As another example, we can have an infinite plane of charge.
We can find solutions to these scenarios without any problems.
However, although we don’t have any paradoxes for an infinite line of charge or an infinite plane of charge,
we nevertheless still have a paradox with an infinite space of charge.
This paradox has not yet been resolved by any of the previous discussions.
There is an analogous paradox that exists for “gravitational” flux when dealing with a uniform “mass” density throughout an infinite space.
But this is a paradox only in Newton’s Theory of Gravity, and not in Einstein’s Theory of gravity.
General Relativity does not have any problems dealing with a uniform mass density throughout all space.
This may be an indication that we also need a better theory of electromagnetism,
so as to be able to handle a uniform charge density throughout all space.
However, there is a major difference between electromagnetism and gravity.
Whereas we have negative electric charges, we do not have negative masses.
Electric charges can cancel each other out, whereas gravitational masses never can.
The Universe must have an overall positive mass density,
but the Universe does not need to have an overall positive charge density.
The answer to the paradox might simply be that it is not physically possible to have a Universe with a net charge density throughout all space.
If a type of Universe is not physically possible, then we might be forbidden from asking what would happen if such a Universe did exist.
Much more information is available in the other videos on this channel.
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