Dimensional Formula

Dimensional Formula

Dimensional formula for some other physical quantities:
Charge, (q) = [AT]
Specific heat, (s) = [L²T²K^-1]
Gas constant, [R] = [ML²T^-2K^-1 mol^-1]

Rules for writing dimensions of a physical quantity
We follow certain rules while expressing a physical quantity in terms of dimensions, they are as follows:

-     Dimensions are always enclosed in [ ] brackets
-    If the body is independent of any fundamental quantity, we take its power to be 0
-    When the dimensions are simplified we put all the fundamental quantities with their respective power in single [  ] brackets, for ex: as in velocity         we write [L][T]^-1 as [LT^-1].
-    We always try to get derived quantities in terms of fundamental quantities while writing a dimension.
-    Laws of exponents are used while writing dimensional formula of physical quantity so basic requirement is a must thing
-    If the dimension is written as it is we take its power to be 1, which is an understood thing
-    Plane angle and Solid angle are dimensionless quantity, that is they are independent of fundamental quantities

Dimensional Formula for different physical quantities:
    It consumes a lot of time while deriving dimensions of quantities. So, in order to save time, we learn some basic dimensions of certain quantities like velocity, acceleration, and other related derived quantities. 
    For Example, suppose you’re asked to find dimensions of Force and you remember dimension of acceleration is [LT-2], you can easily state that the dimension of force as [MLT^-2] as force is the product of mass and acceleration of a body.

IMPORTANT DIMENSIONAL FORMULAE

Dimensional Constants:
The physical quantities which have dimensions and have a fixed value are called as dimensional constants.
Ex: Gravitational Constant (G), Planck’s Constant (h), Universal gas constant (R), Velocity of light in vacuum (c) etc.,
Dimensional variables:
Dimensional variables are those physical quantities which have dimensions and do not have fixed value.
Ex:  Velocity, acceleration, force, work, power... etc
Dimensionless constants:
Dimensionless quantities are those which do not have dimensions but have a fixed value.
(a) Dimensionless quantities without units. 
 Ex: Pure numbers, pi(p) etc.,
(b) Dimensionless quantities with units.
 Ex: Angular displacement - radian, Joule’s constant-joule/calorie, etc.,
Dimensionless variables: 
Dimensionless variables are those physical quantities which do not have dimensions and do not have fixed value.
Ex: Specific gravity, refractive index, Coefficient of friction, Poisson’s Ratio etc.,
Advantages of Dimensions:
Advantages of describing a physical quantity in terms of its dimensions are as follows:
-    Describing dimensions help in understanding the relation between physical quantities and its dependence on base or fundamental quantities, that is how physical quantity rely on mass, time, length, temperature etc.
-    Dimensions are used in dimensional analysis, where we use them to convert from one system to another.
-    Dimensions are used in predicting unknown formulae by just studying how a certain physical quantity depends on base quantities and up to which extent.
-    It makes measurement and study of physical quantities easier.
-    We are able to identify or observe a quantity just because of its dimensions.
-    Dimensions define physical quantities & their existence.
Limitations of Dimensions:
-    Besides being a useful quantity, there are many limitations of dimensions, which are as follows:
-    Dimensions can’t be used for trigonometric and exponential functions.
-    Dimensions never define exact form of a relation.
-    We can’t find values of certain constants in physical relations with the help of dimensions.
-    A dimensionally correct equation may not be the correct equation always.