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 ε0 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.

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Now, let us place a 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.

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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).

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

Types of Permittivity:

Permittivity of a substance

Permittivity of free space

Relative Permittivity

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. Permittivity of a space defines the capability of a substance to allow the electric lines of force to pass through it. Incase of material medium the permittivity varies because different medium has different potential of permitting the electrical field through itself but, incase of free space the permittivity doesn’t vary because it contains no matter or field. Therefore, 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.

\[F \propto \frac{q_{1}q_{2}}{r_{2}}\]

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

\[F = k \frac{q_{1}{q_{2}}}{r_{2}}\]

Now here the constant ‘k’ is \[\frac{1}{4\pi \epsilon_{0}}\]

\[F = \frac{1}{4\pi \epsilon_{0}}\times \frac{q_{1}{q_{2}}}{r_{2}}\]

Where \[\epsilon_{0}\] is the permittivity of free space.

Derivation of permittivity of free space \[\epsilon_{0}\] from coulomb’s law.

If,

\[F = \frac{1}{4\pi \epsilon_{0}}\times \frac{q_{1}q_{2}}{r_{2}}\]

Then,

\[\epsilon_{0} = \frac{1}{4\pi F}\times \frac{q_{1}q_{2}}{r_{2}}\]

From Capacitance:

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

C = 4Πε0R

Thus, ε0 = C/4ΠR

Dimension and Unit of Permittivity of Free Space (ε∘).

1. From Coulomb’s Law,

\[\epsilon_{0} = \frac{1}{4\pi F}\times \frac{q_{1}q_{2}}{r_{2}}\]

\[\epsilon_{0} = \frac{1}{M^{1}L^{3}T^{‘2}}\times \frac{(AT^{1})^{2}}{L^{2}}\]

= [\[M^{1} L^{3} T'^{2} A^{2}]\]

Dimension: [\[M^{1} L^{3} T'^{2} A^{2}]\]

According to the aforementioned dimension, the S.I unit of ε∘ is C²N m². Where C stands for Coulomb, N stands for Newton And m stands for metre.

Unit: C²/Nm²

2. From Capacitance

ε0 = C/4ΠR

As shown above the capacitance of spherical conductor C is 4πrε0. Where r radius of sphere and e is the Permittivity of free space.

The unit of C is farad and its dimensional formula is [M-1 L-2 T4A2]

So ε0=C/r

Where the dimension of r is L.

By solving using capacitance formula

C = Q/V

= Q × Q/W

= Q²/W

= I²t²/W

[C] = [M¹L²T²][I²T²]

= [M¹L²T⁴I²]

ε0= [\[M^{-1}L^{-3}T^{4}A^{2}]\]

Unit: F/m, where Faraday is the unit of Coulomb and metre is the unit of radius.

Value of Permittivity of Free Space:

The value of epsilon naught ε0 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 square per Newton metre square. Coulomb is the S.I unit of charge, Newton is S.I unit of force and metre is S.I unit of length.

Uses of Permittivity of Free Space

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.