Epsilon Naught Value

Epsilon Naught is the permittivity of free space which is also commonly known as vacuum permittivity or electric constant. Epsilon naught is represented by the symbol 0 which is nothing but a Greek alphabet. There is one more similar term and that is epsilon which is denoted by ε. Epsilon is the unit of the permittivity of an insulating, or dielectric material. 

What is Permittivity?

Let us take a positively charged particle and suspend it in free space (vacuum). As we know the direction of electric lines of forces is outward for positive charges and inwards for negative charges. Therefore, the direction of the lines of forces of the electric field will be outwards making equal solid angles between them.

Now, let us place material in free space. The electric field lines will pass through this material more easily as compared to the free space. In this case, we can say that the permittivity of the material is more as compared to the permittivity of free space.

Electric field lines are mosquitoes and permittivity is just like a mosquito repellent in a room that obstructs the entrance of mosquitoes that try to enter the room. It is something that characterises the ability of the atomic charge in an insulating material to distort in the presence of an electric field that attempts to pass through it.

On the other hand, the permittivity of free space represents the capability of a vacuum to permit electric fields. The permittivity of free space is denoted by ε0 or epsilon-naught that is also connected to the energy stored within an electric field and capacitance.

Please note that when we place dielectrics in an electric field, practically no current flows in them because, unlike metals, they have no free electrons that may shift through the material. Instead, electric polarization occurs. Thus, permittivity is the ability of a substance to allow the electric field to pass through it.

Unit of Permittivity: The standard SI unit for permittivity is farad per meter (F/m or F·m−1).

\[\frac{F}{m}=\frac{C}{V.m}=\frac{C^2}{N.m^2}=\frac{A^2.s^4}{kg.m^3}=\frac{N}{V^2}\]

Derivation for the Unit and Dimension of Epsilon-Naught

If we have to find the unit and dimension of permittivity of free space, we can use the above formula.

F =\[\frac{1}{4\pi \epsilon _0}.\frac{q_1q_2}{r^2}\] or ε = \[\frac{q_1q_2}{F4\pi r^2}\]….(1)

We know that unit of force is N or Kgms⁻¹

Distance = meter (m)

Charges q1 q2 = C² = A²s²

4, π is a constant value. 

Then, the unit of permittivity of free space is:

\[\frac{A^2s^2}{Kgms^{-2}m^2}=A^2s^4kg^{-1}m^{-3}\]

We know that 1 Farad or F =s⁴ A² kg⁻¹ m⁻² 

\[\Rightarrow A^2s^4kg^{-1}m^{-3}=Fm^{-1}\].

Here, we converted one system of the unit, i.e., s⁴ A² kg⁻¹ m⁻² to F.

So, the unit of permittivity is F/m.

Now, let’s find out the dimension of permittivity of free space.

We know that the dimensional formula of F =\[[MLT^{-2}]\]  

q1 and q2 = Current * time 

So, the dimension of q * q =\[[A^2T^2]\]

∵ 4, and π are constant values, so their dimension will be \[[M^0L^0T^0]\]

Here, we know one thing that 4, and π are dimensionless constants because these constants don’t possess dimensions.

The dimension of r²= \[[L^2]\]

Now, putting these values in equation (1), we get:

=  \[\frac{[A^2T^2]}{[MLT^{-2}]*[M^0L^0T^0]*[L^2]}=[M^{-1}A^2T^4L^{-3}]\]

This is the dimension formula for the permittivity of free space.

Please note that At a minimum (in a vacuum), ε → εo

This means ε (Epsilon) becomes εo (Epsilon-naught).

The minimum value for the permittivity of free space is 8.85x10⁻¹²C²N⁻¹m⁻²

We know that permittivity is a physical quantity that has a constant value. 

We know that the quantities that have a constant value and possess dimensions are known as dimensional constants. So, permittivity is also a dimensional constant.

Now, let’s find out the unit and dimension of permittivity of free space.

1. Unit of permittivity of free space

When ε → εo, then the formula becomes εo = \[\frac{q_1q_2}{F4\pi r^2}\]

You can see that the formula for permittivity and permittivity of free space is the Difference Between Relative Permittivity and Permittivity of Free Space Same.

So, the unit of permittivity of free space =A²s⁴kg⁻¹m⁻³ or Fm⁻¹ .

2. The dimension of permittivity of free space

Having the same formula as permittivity, the dimension of permittivity of vacuum is also \[[M^{-1}A^2T^4L^{-3}]\].

Types of Permittivity:

• The permittivity of a substance

• The permittivity of free space

• Relative Permittivity

The permittivity of Free Space

Epsilon Naught is used to represent permittivity of free space where free space is a space unoccupied by matter or any field be it electrical, magnetical, electromagnetic or gravitational field. Just like how every medium has its own permittivity even free space has its own permittivity. The permittivity of space defines the capability of a substance to allow the electric lines of force to pass through it. In the case of a material medium, the permittivity varies because the different medium has a different potential for permitting the electrical field through themselves but, incase of free space the permittivity doesn’t vary because it contains no matter or field. Therefore, the permittivity of free space is constant and has a standard value. 

The Formula of Permittivity of Free Space:

From Coulomb’s Law

Coulomb’s law helps us to find the force between two charged bodies. It is directly proportional to the product of the charged bodies and inversely proportional to the square of the distance between them.

\[F\propto \frac{q_1q_2}{r^2}\]

To remove the proportionality we use a constant ‘k’.

\[F=k\frac{q_1q_2}{r^2}\]

Now here the constant ‘k’ is  \[\frac{1}{4\pi \epsilon _0}\]

\[F=\frac{1}{4\pi \epsilon _0}\frac{q_1q_2}{r_2}\]

Where 0

is the permittivity of free space.

Derivation of permittivity of free space 0

 from Coulomb's law.

If,

\[F=\frac{1}{4\pi \epsilon _0}*\frac{q_1q_2}{r^2}\]

Then,

\[\epsilon _0=\frac{1}{4\pi \epsilon _0}*\frac{q_1q_2}{r^2}\]

From Capacitance:

We know that the capacitance of a conducting sphere of radius R is 

C = \[4\pi \epsilon _0R\] 

Thus,   \[\epsilon _0=\frac{C}{4\pi R}\] 

Value of Permittivity of Free Space

The value of epsilon naught 0 is  8.854187817 × 10⁻¹². F.m⁻¹ (In SI Unit), where the unit is farads per meter. Farad is the SI unit of electrical capacitance, equal to the capacitance of a capacitor in which one coulomb of charge causes a potential difference of one volt.

Or The value of epsilon naught is 8.854187817 × 10⁻¹² C²/N.m² (In CGS units), where the unit is Coulomb squared per Newton metre square. Coulomb is the S.I unit of charge, Newton is the S.I unit of force and the metre is the S.I unit of length.

Uses of Permittivity of Free Space

• The permittivity of space as discussed previously is a constant, so it is used in a lot of numerical problems.

• It is used to find the force between two charged particles using Coulomb's law.

• It is used to find the capacitance of an insulator.