Dimensional Formula of Relative Permittivity

Permittivity is a property observed in almost every material to measure the opposition or resistance offered to a body against the formation of an electric field. It gives us the number of charges needed to produce one unit of electric flux in the given medium. This type of opposition is observed in dielectrics as well. The least possible value of permittivity is that of free space or vacuum. In this article, relative permittivity, its formula, dimensional formula and its limitations have been discussed.

Dimensions

Dimensions of a physical quantity are the powers to which the base quantities are raised to represent that quantity. 

Dimensional Formula 

The dimensional formula of any physical quantity is that expression that represents how and which of the base quantities are included in that quantity. 

It is written by enclosing the symbols for base quantities with appropriate power in square brackets, i.e.,

Eg: Dimension formula of mass is: M

Dimensional Equation 

The equation obtained by equating a physical quantity with its dimensional formula is called a dimensional equation. 

Application of Dimensional Analysis 

1. To Convert a Physical Quantity from One System of the Unit to Another: 

It is based on the fact that the magnitude of a physical quantity remains the same, no matter whatever system is used for measurement, i.e., magnitude = numeric value(n) multiplied by unit (u) = constant, 

n₁u₁= n₂u₂   

2. To Check Dimensional Correctness of a Given Physical Relation: 

If in a given relation, the terms of both sides have the same dimensions, then the equation is dimensionally correct. This concept is best known as the principle of homogeneity of dimensions. 

3. To Derive the Relationship Between Various Physical Quantities: 

Using the principle of homogeneity of dimension new relations among physical quantities can be derived if the dependent quantities are known. 

Limitation of this Method

1. This method can be used only if the dependency is of multiplication type. Any formulas containing exponential, trigonometric, and logarithmic functions can not be derived using this method. A formula containing more than one term which is added or subtracted likes \[ s = u_{t} + \frac{1}{2} at^{2}\] also cannot be derived.

2. The relation derived from this method gives no information about the dimensionless constants. 

Relative Permittivity

It is the factor by which the electric field between the charges decreases in comparison or relative to vacuum. Relative permittivity is the ratio of the capacitance of a capacitor using that material as a dielectric, compared with a similar capacitor that has a vacuum as its dielectric.

Whereas dielectric constant is defined as the relative permittivity for a substance or material.

Although these terms may be seen to be related, it is often important to use the correct terminology wherever required.

The relative permittivity of a material is defined as the ratio of its absolute permittivity with respect to the absolute permittivity of a vacuum. It is usually defined as a dielectric medium relative to the permittivity of a vacuum or free space. If the permittivity of a medium is given by ϵ then the relative permittivity is given by

ϵ_r = ϵ / ϵ₀ where ϵ₀ is the permittivity of vacuum. The relative permittivity is a unitless quantity as it is a ratio of two permittivity values. 

When designing capacitors, the information on relative permittivity is useful or in a situation where material might produce some capacitance into a circuit.

The Formula of Relative Permittivity

Relative permittivity is written as

\[\epsilon\] = D/E

Here, \[\epsilon\] = permittivity of the substance in Farads per metre

D = electric flux density

E = electric field strength

From the above definition of permittivity, it is clear that the constants are related according to the following equation:

\[\epsilon = \frac{\epsilon_{S}}{\epsilon_{0}}\] Here, \[\epsilon_{r}\] = relative permittivity

\[\epsilon_{S}\] permittivity of the given substance, and, \[\epsilon_{0}\] = permittivity of a vacuum, and both of them are measured in terms of  Farads per metre.

Dimensional Formula of Relative Permittivity

The dimensional formula of Permittivity of free space or vacuum is written as M¹ L^-3 T⁴ I² where M stands for mass, L stands for length, T stands for time, and I stands for a charge.

Derivation of the Dimensional Formula of Relative Permittivity

As it is known that, Force =4π\[\epsilon\]^-1 × [Charge²] × Distance^-2

Therefore, Permittivity (\[\epsilon\]) = [Charge²] × Force^-1 × Distance^-2 . . . . . (1)

As, Charge = current × time,

So, the dimensional formula of charge = [I¹ T¹] . . . . . (2)

And, the dimensional formula of distance = [M⁰ L¹ T⁰] . . . . . (3)

As, Force = Mass × Acceleration,

So, we can write the dimensional formula of force as = [M¹ L¹ T^-2] . . . . . (4)

On putting equations (2), (3) and (4) in equation (1) we get,

Permittivity = Charge² × Force^-1 × Distance^-2

or, \[\epsilon\] = [I¹ T¹]² × [M¹ L¹ T^-2]^-1 × [M⁰ L¹ T⁰]^-2 = [M¹ L^-3 T⁴ I²]

Therefore, the permittivity of free space or vacuum is dimensionally represented as M¹ L^-3 T⁴ I².

So, from the above equation, we derived the dimensional formula of permittivity.

SI Unit of Permittivity

The unit of permittivity is C²N^-1m^-2. The steps for finding the formula of the unit of permittivity are as follows:

• First of all, the force F between two charges (q₁ and q₂) have to be calculated by using Coulomb's law when permittivity is ϵ0. Therefore,

 F = q₁.q₂/(4πϵ₀r²)

=> ϵ₀ = q₁.q₂/(4π.F.r²)

• Then all the SI units should be put for the above equation in the RHS and we will get the unit of permittivity as ϵ₀ = C.C/N.m² which equals C²N^-1m^-2. C²/N is also called Farad. Therefore, the SI unit of permittivity is Farad m^-1.

The value of permittivity of free space is 8.85 × 10^-12 Farad/meter.

Dimensional Formula of Relative Permittivity

The dimensional formula of relative permittivity of vacuum is M¹ L^-3 T^-4 I².

Applications

The concept of relative permittivity is used in dielectric materials which are non-conductors of electricity but possess the ability to be highly polarized by an electric field. Such dielectric materials have many applications, some of which are enumerated as follows:

• Electronic components are responsible for the energy storage properties of a device such as a capacitor.

• Devices with high or low permittivity or dielectric constant are widely used in semiconductors that enhance their performance and reduce the size of the device as well.

• Dielectric materials are used in devices used in display devices such as LCDs.

• Piezoelectric MEMs used in micropumps for inkjet heads or micro-gyrosensors are also dielectric materials.

• Some extent of dielectric properties is observed in ceramics and polymers.

• Dielectric materials are also used in the textile industry, electronic sewing machines, and food processing industries.

• They are used in radio frequency transmission lines, optical fibres are a good example of such a communication line system.

• Relative permittivity can be used in chemistry to measure a solvent's chemical polarity. This information is important in techniques in analytical chemistry.

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