Dimensional Formula of Relative Permittivity

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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 which 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., [ ].

E.g: 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 = ut+ ½ at2 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 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.


Formula of Relative Permittivity

Relative permittivity is written as

 ε = D/E

 Here, ε = 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:

εᵣ=   ε\[_{s}\]/ε₀

Here, εᵣ = relative permittivity

ε\[_{s}\] = permittivity of the given substance, and, ε₀ = 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 = [4πε]-1 × [Charge2] × [Distance]–2

Therefore, Permittivity (ε) = [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 equation (2), (3) and (4) in equation (1) we get,

Permittivity = [Charge]2 × [Force]–1 × [Distance]–2

Or, ε = [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.

FAQ (Frequently Asked Questions)

Q1.What Do You Mean By Permittivity?

Answer: Permittivity is defined as the material property that affects the Coulomb force of interaction between two point charges in the material. Likewise, the relative permittivity of a medium is the ratio of the capacitance of a capacitor using that material as a dielectric when compared to a similar capacitor that has vacuum as its dielectric.

Q2.What is the Formula for Relative Permittivity?

Answer: C = εᵣ C₀ where C₀ ≡ ε₀ A / d. 

εᵣ  = relative permittivity. The definition of the term relative permittivity is commonly given as follows.

Relative permittivity is equal to the ratio of the capacitance of a capacitor when filled with a dielectric to the capacitance of a similar capacitor with vacuum, without any dielectric material.

Q3. What Happens to the Value of Relative Permittivity When the Actual Permittivity Decreases?

Answer: We know that the actual relative permittivity is the ratio of actual permittivity to the absolute permittivity. Thus, it is directly proportional to actual permittivity. So, as the actual permittivity decreases, the relative permittivity of the medium also decreases.