Principle of Homogeneity of Dimensions for JEE

Before moving on to the principle of homogeneity of dimensions, we’ll first look at the basics of units and dimensions. In Physics, the process of measurement is fundamental in order to compare different physical quantities on a daily basis. Now to measure certain standard quantities like the mass, temperature, length, etc. which are measured worldwide, certain standard chosen units are accepted and they are used worldwide. 

There are certain fundamental quantities and the other quantities can be derived from these fundamental quantities and they are known as derived quantities. Each derived quantity must be represented by the appropriate power of the corresponding fundamental quantity. Dimensions are the powers of fundamental quantities that allow them to be elevated to represent unit derived quantities. In other words, the powers to which the fundamental quantities (base quantities) are increased to represent a physical quantity are its dimensions. We’ll look at it in more detail.

What are Base Quantities?

Now there are certain fundamental physical quantities and all the other physical quantities can be derived using these fundamental physical quantities. The fundamental quantities include the mass, length, time, temperature, current, luminous intensity, and amount of substance, radian and steradian. These can be called the base quantities. The power to which the base quantities are elevated to represent a physical quantity is its dimensions. Any physical quantity's dimensional formula is an expression that shows how and which base quantities are used to create that quantity. The symbols for the basic values are put inside square brackets with the appropriate power.

There are many different systems of units like the MKS, CGS, FPS and the standard SI system. We won’t go into the details of this but just for knowledge MKS stands for Metre Kilogram Second, CMS stands for Centimetre Gram Seconds, FPS stands for Foot Pound Second and SI stands for Système international or the International System of Units.

Mathematical Representation of Dimensions

Now that we know what dimensions are, we’ll see how to represent them. This mathematical representation of dimensions is known as the dimensional formula. The dimensional formula is just an expression that shows the power to which the fundamental units are raised in order to obtain one unit of the derived quantity.

A physical quantity's dimensional formula is denoted by $[M^aL^bT^c]$ when it depends on the basic dimensions M (Mass), L (Length), and T (Time) with respective powers a, b, and c. The other dimensions are temperature $[K]$, charge $[Q]$, luminous intensity $[Cd]$, and amount of material $[mol]$. Radians and Steradians are dimensionless quantities as they measure angles. 

Dimensions are always enclosed in square brackets. We consider the power to be zero if the physical quantity is independent of any basic quantity. For example the dimensional formula of length is $[M^0L^1T^0]$. Similarly, the dimensional formula of time will be $[M^0L^0T^1]$ and the dimensional formula of mass will be $[M^1L^0T^0]$. 

The speed v is given as

$\begin{align} &v=\dfrac{\text { distance }}{\text { time }} \\ \\ &v=\dfrac{\left[M^{0} L^{1} T^{0}\right]}{\left[M^{0} L^{0} T^{1}\right]} \\ \\ &v=\left[M^{0} L^{1} T^{-1}\right] \end{align}$

The dimensional formula of velocity and speed will be the same as they are basically the same physical quantity. The dimensional formula of speed is thus $[M^0L^1T^{–1}]$.

If we again divide the velocity by time, we’ll get the dimensional formula of acceleration a.

$\begin{align} &a=\dfrac{\text { velocity }}{\text { time }} \\ \\ &a=\dfrac{\left[M^{0} L^{1} T^{-1}\right]}{\left[M^{0} L^{0} T^{1}\right]} \\ \\ &a=\left[M^{0} L^{1} T^{-2}\right] \end{align}$

So $[M^0L^1T^{-2} ]$ is the dimensional formula of acceleration. 

Force F equals mass time acceleration, so the dimensional formula of force can be found as

$\begin{align} &F=\text { mass } \times \text { acceleration } \\ \\ &F=\left[M^{1} L^{0} T^{0}\right] \times\left[M^{0} L^{1} T^{-2}\right] \\ \\ &F=\left[M^{1} L^{1} T^{-2}\right] \end{align}$

The dimensional formula for work W can be found by the product of force and displacement.

W = Force x displacement

\[ W=\left[M^{1} L^{1} T^{-2}\right] \times\left[M^{0} L^{1} T^{0}\right] \]

\[ W=\left[M^{1} L^{2} T^{-2}\right] \]

The dimensional formula of work is $[M^1L^2T^{-2}]$.

Power dimensional formula P can be found by dividing the work done by time.

P = \[ \frac{Work}{time} \]

P = \[ \dfrac{\left[M^{1} L^{2} T^{-2}\right]}{\left[M^{0} L^{0} T^{1}\right]} \]

P = \[ \left[M^{1} L^{2} T^{-3}\right] \]

We can also find out the dimensional formula of current A.

Now current is equal to the charge per unit time. Charge has a dimensional formula $[M^0L^0T^0Q^1]$. 

So, current dimensional formula will be

$\begin{align} &A=\dfrac{\text { Charge }}{\text { time }} \\ \\ &A=\dfrac{\left[M^{0} L^{0} T^{0} Q^{1}\right]}{\left[M^{0} L^{0} T^{\prime}\right]} \\ \\ &A=\left[M^{0} L^{0} T^{-1} Q^{1}\right] \end{align}$

These were some basic dimensional formulas.

Principle of Homogeneity of Dimensions

According to the principle of homogeneity of dimensions, each term in a dimensional equation should have the same dimensions on both sides. The foundation of dimensional homogeneity is the ability to add, subtract, or compare only physical amounts of the same kind. Now this idea is readily used in dimensional analysis.

Utilising a set of units to identify an equation's form or, more frequently, to verify the accuracy of a computation's output is known as dimensional analysis. Most physical quantities require a unit to be presented in order to be represented numerically. Not all quantities, though, call for their own unit. Using physical principles, units of quantity can be expressed as combinations of units of other quantities.

Therefore, just a small number of units are needed. These are referred to as basic or fundamental units, whereas others are referred to as derived units. Since they may be expressed in terms of fundamental units, derived units are helpful. The basic fundamental units are metre (m), second (s), kilogram (kg), Ampere (A), and Kelvin (K). 

The foundation of dimensional analysis is the assumption that the variables in a physical phenomena should be properly arranged to result in an equation with homogeneous dimensions. An equation is said to be dimensionally homogeneous if the dimensions of the left and right sides are the same.

We can convert units from one type to another thanks to this important notion. Every valid equation must be dimensionally homogeneous, which calls for all additive terms to have the same size on all sides of the equation.

Uses of Dimensional Analysis

• It is used to prove the validity of a homogeneity-based equation or other physical relationship. Both sides of the equation should take dimensions into consideration. The validity of the dimensional relation depends on whether the LHS and RHS of an equation have equal dimensions. The relationships will be inaccurate if the measurements on two sides are off.

• To convert the value of a physical quantity from one unit system to another, dimensional analysis is utilised.

• Understanding the relationship between physical quantities and their dependency on base or fundamental quantities, such as how a body's dimensions depend on mass, time, length, temperature, etc., is made easier by describing its dimensions and using dimensional analysis.

• By just examining how and to what extent a particular body depends on foundation quantities, dimensions are utilised to forecast unknowable formulas.

Table of Dimensions

Conclusion

We select certain units that are widely used to measure each of the standard physical quantities. These units can be used to express and measure a variety of other similar amounts. Fundamental quantities are called base quantities and the equation that describes how and which base quantities are employed to produce any physical quantity is known as the dimensional formula. 

Square brackets with proper power are used to surround the symbols for the fundamental values. Each term in a dimensional equation should have the same dimensions on both sides, according to the homogeneity of dimensions principle. Dimensional analysis is used to change units, check the correctness of any equation and also to predict the formula of unknown quantities based on the relationship between the unknown quantities and the base quantities.