Formulas and Equations Related to Dimensions
Energy is the ability to do work. In other words, it is the capacity of doing work. In Physics, energy is a quantitative property that is transferred to the object by the object in order to perform work in the form of heat or any other form as per desired results. Various common forms of energy are kinetic energy, potential energy, thermal energy, elastic energy, gravitational energy, magnetic energy, radiant energy, etc. All these forms of energy are interconvertible from one form to another depending on the parameters available for these conversions.
The SI unit of energy is joule and can be defined as the amount of work done to move an object by applying the force 1N to the distance of 1 metre in the same direction of applied force.
An equation with dimensions is one that describes derived units in terms of fundamental units. Metric units are metre, kilogram, second, ampere, kelvin, mole, and candela, with length, mass, time, electric current, temperature, number of particles and light intensity taken as seven fundamental quantities. When an equation is written, a dimensional formula is used to determine how individual quantities relate to each other. The following example illustrates what a dimensional equation looks like.
The dimensions of an area can be described by the following equation:
An area is calculated by multiplying the length by the width
The length multiplied with the length
\[[L] * [L]\]
\[ = [L]^{2} \Rightarrow \, Dimensional \, formula \, (equation) \, for \, area (A) = L^{2} M^{0} T^{0}\]
Dimensional Formula of Energy
The energy possessed by the body or transferred to or by the body is expressed in terms of a joule. The unit of energy is named after James Prescott Joule because he independently discovered the mechanical equivalent in a series of experiments using his famous Joule apparatus which comprises a descending weight attached to a string, causing rotation of a paddle immersed in water and was insulated from heat transfer.
The apparatus showed that the gravitational potential energy lost by the weight while descending was equal to the internal energy gained by the water through friction with the paddle. Thus it states that energy change or work is done in applying a force of one newton through a distance of one metre and establishing energy equivalence with the work.
Numerically, Energy is equivalent to work done. Thus mathematically
Energy =Work Done=Force*displacement
SI unit of force = Newton
1 Newton= \[1kgms^{-2}\]
SI unit of displacement = metre
Thus, 1 Joule=\[1kgm^{2}s^{-2}\]
Dimension of mass =M
Dimension of length=L
Dimension of time =T
Using the above derivation 1 Joule=\[1kgm^{2}s^{-2}\]
We can derive the dimension of energy =\[ML^{2}T^{-2}\]
An Alternate Method to Derive the Unit of Energy
Using Einstein’s famous equation of mass-energy equivalence, which gives the relation between mass and energy conversion and states that
\[E = mc^{2} \]
Where m= mass and c=velocity
Since mass is measured is kg
Thus the dimension of mass is M
And velocity is the rate of change of position and has unit \[ms^{-1} \]
Therefore the dimension of velocity = \[LT^{-1} \]
Thus dimension formula of energy = \[ML^{2}T^{-2}\]
From both the derivation using E=work done and \[E = mc^{2} \]
We can say that the dimension formula of energy is \[ML^{2}T^{-2}\]
Dimensional Formula of Energy Density
Energy density is defined as the energy stored per unit volume.
Thus mathematically,
Energy Density = Energy/Volume
And, Energy=Force×displacement
Energy=ma×d
Where m=mass,a=acceleration and d=displacement
Thus,dimensions formula of energy = \[M^{1} L^{2} T^{-2}\]
Dimension formula of Volume = \[M^{0} L^{3} T^{0}\]
Using, \[ Energy \, Density = Energy * Volume^{-1} \]
\[ = [M^{1} L^{2} T^{2}] * [M^{0} L^{3} T^{0}]^{-1} = M^{1} L^{-1} T^{-2} \]
Therefore, the dimension formula of energy density is represented as \[M^{1} L^{-1} T^{-2}\]
FAQs on Dimension Formula Of Energy
1. What is the dimension of Surface Energy?
Surface energy can be defined as the energy required to increase the surface area of the liquid by one unit. Thus, the unit of surface energy is \[ Jm^{-2} \]. The dimension of the surface energy is \[M^{1} L^{0} T^{-2}\]. Because of the surface tension of liquids, their free surfaces have a very small area. To increase the area of the surface, it is necessary to bring molecules from the interior to the surface. To do this requires work against the forces of attraction. This work is stored with the molecule's potential energy.
2. What is energy and what is the dimension for Potential Energy?
In physics, energy is defined as the capacity of doing work. Now the expression for the potential energy is. P.E. = mgh and its dimension are the same as that of energy. The dimension of the potential energy is \[M^{1} L^{2} T^{-2}\].
3. How can we calculate Power from work done?
Power is the measure of the amount of total work done or energy transfer that is done in a given amount of time. Power equals work (J) divided by time (s). Thus the standard unit of power is the watt (W), which equals 1 joule of work per second (J/s). Power can be defined as the rate at which work is done. It is equal to the ratio of work done to the time taken. Mathematically, it can be calculated using the following equation.
Power = Work / Time or P = W / T
4. What are the different forms of Energy?
The different forms of energy are thermal energy, radiant energy, chemical energy, nuclear energy, electrical energy, sound energy, elastic energy, gravitational energy, etc.
5 What applications do Dimensional formulas have?
Following are some applications of the dimensional formula:
An equation's correctness is verified using it.
Different physical quantities can be related with the help of the dimensional formula.
Conversions between different metric systems for any physical quantity can be done using this formula.
By using fundamental units, one quantity can be expressed in terms of the metric system.