Introduction to Dimensions With Different Units and Formula for JEE

A unit is a specific mechanism to associate a numerical value with a quantitative dimension. Currently, there are three main unit systems in use:

• The International System of Units (SI units, from Le Systeme International d’Unites, more commonly simply called metric units)

• The English Engineering System of Units (commonly called English units)

• The British Gravitational System of Units (BG)

Standard Units

We may measure length, height, weight, temperature, mass, and a variety of other quantities using standard units, which are widely used units of measurement. Since these units are standardized, everyone understands the size, weight, and other characteristics of objects and things when measured in these units.

Characteristics of Standard Unit

The following characteristics should be included in every unit chosen to measure a physical quantity.

• It should have a distinct concept and be adequately defined.

• It shouldn't alter in response to changes in external factors like pressure, stress, or temperature.

• The size should be appropriate, neither too big nor too little.

• It shouldn't alter based on the location or time.

• It should be reproducible. 

• It should be internationally accepted.

Classification of Units

Two categories can be used to classify units.

• Fundamental: A physical quantity that does not depend on other physical values. The fundamental quantities of mechanics are mass, length, and time. These fundamental physical quantities are measured in units referred to as fundamental units. For instance, weight, length, time, etc.

Unit and Symbol of Physical Quantities

• Derived: A derived unit is derived from the fundamental units, for example, the unit of area, force, speed, etc.

Supplementary Units

• Radian (rad): It is a supplementary unit for plane angles. It is the plane angle that a circle's radius-equivalent arc subtends at the center of the circle. It is denoted by $\theta $. 

$\theta =l/r$ ; 𝑙 is length of the arcand 𝑟 is radius of the circle

• Steradian (Sr): It is an additional unit for solid angle. It is the angle that a sphere's surface area with a magnitude equal to the square of the sphere's radius forms at its center. It is denoted by Ω.

$\Omega =\Delta s/{{r}^{2}}$

Dimensions

The unit of measurement used to express a physical variable quantitatively is called a dimension. The fundamental units of mass, length, and time, denoted by the letters M, L, and T, are raised to powers that consider both their nature and not just their magnitude. For example,

• Area = Length x Breadth

• Area= $[{{L}^{1}}]\times [{{L}^{1}}]=[{{L}^{2}}]=[{{M}^{0}}{{L}^{2}}{{T}^{0}}]$

The dimensions of area in mass, length, and time are designated by the power (0, 2, 0) of the fundamental units, respectively.

Dimensions and units are distinct concepts. A unit is a technique to give a number or measurement to that dimension, whereas a dimension is a measure of a physical variable (without numerical values). For instance, length is a dimension but is expressed in units of meters or feet (ft) (m).

Dimensional Formula

An expression that describes how physical amount depends on fundamental physical quantity, together with the powers of mass, length, and time. Example,

Speed = Distance/Time

Speed$=[{{L}^{1}}]/[{{T}^{1}}]=[{{M}^{0}}{{L}^{1}}{{T}^{-1}}]$ 

It explains that L & T have an impact on speed. It is independent of M. Refer to the Dimension Formula PDF for a more detailed understanding.

Dimensional Equation

A dimensional equation is one that is created by combining the physical quantity and its dimensional formula. The following is the area dimensional equation:

Area = $[{{M}^{0}}{{L}^{2}}{{T}^{0}}]$

Physical Quantities

Physical quantities refer to any quantities that can be measured and in which the principles of Physics can be stated. For instance, consider distance, speed, mass, and force.

Dimensional Formula for All Physical Quantities

Dimensional Formula of Other Quantities

• Dimension of Light Year: [M⁰L¹T⁰]

• Dimensional formula for Coefficient of Self-Induction: [M¹L²T^-2A^-2]

• Force constant dimensional formula: [M¹L⁰T^-2]

• Dimensions of Permittivity of Free Space: [M^-1L^-3T⁴I²]

• Dimension of Stefan Constant: [M¹L⁰T^-3K^-4]

Conclusion

A few basic quantities or their combinations provide all of the physical quantities. These essential quantities units are known as base units, and combinations are known as derived units. The knowledge of dimensions and units will be beneficial for understanding several advanced concepts of science.