LAW OF HOMOGENEITY OF DIMENSIONS

The given article is about the dimensional analysis of a given equation. We will also see how to perform the basic dimensional analysis. But at the same time, readers should know that according to the international standards there are a total of seven physical quantities. These are the quantities that we will use to form a formula of homogeneity and perform the dimensional analysis. In the dimensional analysis, we will also see how to prove a given formula knowing the quantities on which a quantity needed to be expressed is dependent.

DIMENSIONAL ANALYSIS

Dimensions of the physical quantities are the power to which fundamental units are raised to get the required physical quantity. And the corresponding equation equating the quantity to the powerful combination for its dimensions is called the dimensional formula.

There are 7 fundamental quantities which are the base and the power which gives the dimensional formula. These are –

1. Mass(M)

2. Length(L)

3. Time(T)

4. Electric Current(A)

5. Temperature(K)

6. Luminous Intensity(C)

7. Amount of substance (mol)

They were decided as per the international standards. We can express any equation as the formula of homogeneity and any physical quantity as the dimensional formula using them. But every time we try to deduce the dimensional formula of the given quantity we need the formula for the quantity in terms of either base physical quantity or the derived physical quantity.

E.g.

Volume=L3 ,So the dimensional formula will be

V=[M0L3T0]

Density= Mass/Volume

D=[M1L-3T0] 

Force=ma

F=[MLT-2]

In the same way they are written as the dimensional formula, the corresponding units of the quantities can also be derived by deducing the units of the base fundamental quantities.

Now we will write the units of the physical quantities from their dimensional formula.

V=[M0L3T0]

unit= m3

D=[M1L-3T0] 

unit= kg/m3

Force=ma

F=[MLT-2]

unit=kgm/s2

This unit in respect of Sir Isaac Newton was named Newton(N).

THE LAW OF HOMOGENEITY

The law of homogeneity was proposed to verify the validity of the equations which describe the laws of nature. First of all, we need to define what an equation is. As clear from its name, the equation means two quantities expressed as the combination of multiplication, division, and addition of the many terms equated to each other.

The law of homogeneity says that quantities present there at both sides of the equation should have the same dimension. If you don’t know what dimension is, it is the power to which the fundamental quantities are raised to get the given quantities. It is represented as [MxLyTz]. So the principle of homogeneity of dimensions says that the quantities in addition and the quantities which are equated should always have the same dimensions.

This holds in the same form as the units. The units of the quantities being added should also be always the same. While you CAN equate two different units but the units should be representing the same physical quantities.

DERIVATIONS USING THE FORMULA OF HOMOGENEITY

With some pros and cons, there is a way of deriving formulas using the law of homogeneity. We can deduce a formula of a given quantity if we know what are the quantities on which the given quantity is dependent. The first step to derive is to write the given quantity for which the equation is needed to be derived, as the formula of homogeneity by first writing the dimensional formula of the given quantity. And then writing the quantities with which it varies and putting the variable power to them. Now finally we will equate the corresponding powers from both sides of the equal sign as per the law of homogeneity. But this way we will always face difficulty as there can be any constant factor multiplied by the quantities in terms of which we are expressing the required quantity which doesn’t have any dimension. So we will never be able to figure it out using the principle of homogeneity of dimensions.

E.g. Let’s try to figure out the expression for the period of the simple pendulum which depends on the length of the suspension and the acceleration due to gravity.

 T=kLxgy; where k is any experimentally found constant 

Using the law of homogeneity we can say that the dimensions of both sides of the equation should be equal.

[T]=[L]x[LT-2]y    

Now we can compare

x+y=0, and -2y=1

Which gives y=-½ and x=½ 

So, T=k√l/g 

And this k is experimentally found to be 2𝝿.

CONCLUSION

In this article, we have gone through one of physics’ most fundamental parts which is the law of homogeneity. The readers are advised to give the first chapter of the class 11th physics, that is, Units and Dimensions, a little more time and to make their basic knowledge strong. We have seen the dimensional analysis of a few of the physical quantities. There is an easy way of deriving the complex formulas of physics using the law of homogeneity. But it is also true that to hold the principle of homogeneity of dimensions doesn’t ensure the validity of the expression. This is because the constant factors are impossible to predict that way as they don’t have any dimensions. This way we can say that for an expression to be correct the law of homogeneity is the necessary condition( not sufficient condition ).