Dimensional Formula Of Permittivity

The dimensional formula of any bodily amount is the expression that represents how and which of the bottom portions are protected in that amount. It is denoted by enclosing the symbols for base portions with suitable strength in rectangular brackets, i.e., [ ]

An example is the dimension formula of mass, which is written as [M]. 

The permittivity of free space, represented by ϵ0,  is common terminology used in electromagnetism. It is defined as the capability of a vacuum to permit electric fields. It is used in different formulas like energy stored in an electric field and capacitor, energy density, etc. 

Dimensional formula of permittivity of free space

The dimensional formula for permittivity of free space is given as M-1L-3T4A2 

Where M refers to Mass

            L refers to the length 

            T refers to time and

            A refers to current, with all units in their standard form.

Derivation of the dimensional formula of permittivity

The basic formula that relates permittivity of free space with the electrical force between two charges is : 

      F = (1/4πϵ0).q1.q2/r2

Where F is the electric force between two charges,

        q1, q2 are the charges,

       r is the distance between the charges 

      ϵ0 is the permittivity of free space.

The dimensional formula of force is [MLT-2]

The dimensional formula of charge (q) is [AT]

The dimensional formula of distance ( r ) is [L]

Assembling all the formulas, we get the dimensional formula of 

 ϵ0 is [ M-1L-3T4A2 ]

The dimensional formula for permittivity in free space is the same as the permittivity in any medium. 

Application of permittivity:

Permittivity is useful in Gauss’s law in magnetism, electric field due to infinite wire, Coulomb’s law, and electric field-induced due to the spherical shell.

Gauss’s Law in magnetism

According to Gauss’s law, in electrostatics, the electric charge within the imaginary Gaussian or the closed surface is equal to 1 time of the net electric flux through any closed surface. The Gauss law of electrostatics is written as ∫E⋅dA = Q/ε0.

Where,

• E is the electric field vector
• Q is the enclosed electric charge
• ε0 is the electric permittivity of free space
• A is the outward pointing normal area vector

Electric field due to infinite wire

Consider a wire that is infinitely long and has a linear charge density. Due to the symmetry of the wire, let’s take a cylindrical Gaussian surface, for instance, to compute the electric field. Because the electric field E is radial in direction, flux through the end of the cylindrical surface will be 0 because the electric field and the area vector are perpendicular to each other. The curved Gaussian surface will be the only source of electric flux. The magnitude of the electric field will be constant because it is perpendicular to every point of the curved surface.

E.2πL = λL/ε0

So, E = λ/2πε0x

E = Electric field at a distance x

x = Distance from the axis of the cylinder

λ = charge per unit length

Ε0 = Vacuum permittivity

Coulomb’s law formula

Coulomb’s law allows us to calculate the force between any two charges. According to Coulomb’s law, two charged things will attract or repel each other with a force comparable to the product of their masses. 

Let’s make an equation out of this, shall we?

F= K Q1Q2/d2

K = 14𝝿ε0

Depending on the charges, F is the force of attraction or repulsion.

• ε is absolute permittivity,
• K or εr is the overall permittivity or explicit inductive limit
• ε0 is the permittivity of free space.
• K or εr is likewise called a dielectric steady of the medium where the two charges are set.

Electric field-induced due to the spherical shell

Electric field due to a spherical shell with an inner closed-cell will be quite different from that induced by a standard circular disc. The surface charge density in such a case would be denoted as σ. If the radius of the inner shell is a and the radius of the outer shell is b, then the electric field of this surface can be given as

Ea = q40a2

And 

Eb = q40b2

 ε0 is the permittivity of free space

Hence, we get two different ways to evaluate the electrical field.

• By evaluating the electric field of the inner circle.

• By evaluating the electric field of the outer circle.

Conclusion

The dimensional formula of electrical permittivity of free space can be calculated through the above steps. It is also related to the permittivity of a medium by a dielectric constant “k”. The formula is given as ϵ = ϵ0k. Dimensional analysis has numerous practical applications and can also be used for checking formulas of different quantities.