It is the permittivity of free space. The Epsilon Naught is written as (εo), which is also called Epsilon Zero. It is the spelling-out or sounding-out of the term “εo” — the Greek letter ε, pronounced “epsilon”, and the subscript numeral o, pronounced “naught” (or sometimes “zero” or “sub-zero”).
As others on this thread have mentioned, εo is used in physics to mean the dielectric permittivity of free space (perfect classical vacuum). As of this posting, nobody has pointed out the meaning of εo in mathematics: The smallest ordinal transfinite number is a fixed point of an exponential map, therefore useful in induction proofs.
Learning about the uses of Epsilon Naught Value can also help you understand this concept better.
Introduction to Epsilon Naught Value
In physics, epsilon naught is the permittivity of free space. This is a mathematical quantity that represents the amount of electric field that is permitted in free space or vacuum. Similarly, epsilon of water means how much electric field is permitted in water (or can cross the water). 1/4(pi)(epsilon naught) is 9*10^9.
This number tells us that 9*10^9 field lines are crossing by a charge in a vacuum, but this number may change for water, and the number of field lines penetrating is also changed. Epsilon naught is the permittivity of free space, which is a fundamental constant in electromagnetism. It shows up in Maxwell’s equations and is defined by its relation to the speed of light and the permeability of free space.
The formula for absolute permittivity of free space. It is derived from Columbus’s law. It states that:
Here, F is the force between 2 electric charges. Q1 and Q2 are the two charges, and r is the distance between two charges. If we rearrange the above equation, we will get
Units of Epsilon Naught Value
The Standard Unit of any absolute permittivity is FM-1 (Farad per meter).
Two of the most complex and least-understood constants in Physics are the Permittivity and Permeability constants. It’s always fun watching people try to explain these two. Just what are these constants? How do they affect the bigger picture?
Let’s stop that line of thought for a minute. We’re going to diverge to a physics topic that may seem to be off course(or is it?). Going back to elementary Physics, the Newton-Laplace equation.
Speed of Sound
c is the speed of sound
p is pressure
ρ is density
This is the way to calculate how fast sound waves move through a medium. I’m sure you’ve seen it before. As you increase pressure or decrease density, the speed of sound increases. This can also be changed to be based on temperature since:
T is temperature
R is a gas constant that allows us to convert this between different mediums since the molecules are different sizes and have varying degrees of freedom.
So basically, this is saying temperature increases as pressure increases or density decreases. Good to know!
If we set RR to 11, by setting our units to natural units for the given medium, our above speed of sound equation could be written:
c is the speed of light
ε0 is the vacuum permittivity and spoken epsilon naught
μ0 is the vacuum permeability and spoken mu naught
Okay, now let’s leave this sit a moment, and we’ll return to our original question.
Speed of Light in the Vacuum
We’ll start with an equation that looks oddly similar to our speed of sound equation above:
Where cc is the speed of light
ε0ε0 is the vacuum permittivity and spoken epsilon naught
μ0μ0 is the vacuum permeability and spoken mu naught
Let us now talk a bit about history. Human history is full of examples of people overcomplicating very simple things. Ever find the US Tax Code of Federal Regulations and Administrative Rules in a library? It takes up an entire section. Or go to a remote village in the Laotian rainforests to hear their elaborate stories to explain everything from the Sun to why different traditions exist. We constantly are making up more of things than there is.
And that brings us back to the permittivity and permeability of free space.
We’re not going to discuss the history of these or how we found them. You can read all about that on Wikipedia. We are only going to simplify them to what they truly represent.
The permittivity of Free Space is-
s are seconds
A are Amperes
m are meters
kg is kilograms
What is an Ampere?
There are two components of the Ampere that are at a right angle to each other. One of the most important components is the amount of current moving through two wires a meter apart. The other component is the force between the two wires, which is perpendicular to the current. You should also know that each wire has 1 Ampere of power going through them. We’re going to do something a little unconventional and use the force component between the two wires.
But we will need to divide this by two because 1 Ampere is going through each wire.
N are Newtons
So what’s a Newton?
So this whole time, we could have just used kilograms, meters, and seconds. Let’s try this again but with just these 3 units:
So this is just kilograms divided by meters cubed. That happens to be density(m/V). It’s the density of the vacuum, the medium which carries light waves, which implies space has a tiny bit of mass. Let’s change this to:
Onto permeability now:
And using the orthogonal value of the Ampere:
When we measure pressure, we are measuring force divided by an area.
As you can see above, the only difference between pressure and our inverted permeability is that pressure has another unit of length in both the top and bottom. In other words, it is the same as pressure. But if we multiply back in another meter unit to the top and bottom, we get Energy density:
p=F/Area=N/m2= kgms2= kgm2s2
That is just the quantity of kinetic energy per cubic meter, which makes sense. The more energetic a gas is, the more pressure there is.
If we look back at our equation for how these relate to the speed of light, we could rewrite this part as:
It’s just inverse pressure. Let’s change this to:
p= kgm2s2 ÷ m3= Jm2
Which is the same as 1/μ01/μ0. So the permeability of space is just the (inverse)pressure of space, which is the same as the kinetic energy of space per cubic meter(E/V). That sure is a lot more simple to understand, at least for me!
Therefore the speed of light equation becomes:
Let’s rewrite it like this:
C = EV/m/V
VV is Volume (i.e. m3m3)
EE is Energy
mm is mass
To clarify this, it’s referring to space itself as a substance transmitting vibrational waves, which we see as light, matter-waves, neutrinos, etc. So the speed of light is equal to the square root of the kinetic energy per volume divided by the mass per volume.
If we want to take that a step further, we can cancel the volume on the top and bottom of the fraction which gives us back:
C = E/m
And moving things around, we get:
By now, you will be able to understand the concept of epsilon value. You know the definition, the formulas, and how it is calculated. For a deeper understanding, you will have to go through the chapter thoroughly.