Dimensional Formula of Energy Density

Energy density is a concept used in a variety of branches in physics. Energy is associated with various systems such as chemical, electrical, and nuclear physics, and to have a generalised idea about the energy content of the system, the concept of energy density is applied. In the fields of astronomy, astrophysics, and cosmology, energy density has many applications. In these notes of the dimension formula for energy density, we discuss the definition and the derivation for its dimensional formula.  

Dimensional Formula of Energy Density

From the definition of energy density, we know that

Energy density (E) = energy/volume

So, the dimension of energy density = dimension of energy / dimension of volume.

Dimension of energy = dimension of work done = dimension of force x dimension of displacement 

 = [MLT-2].[L] = [ML2T-2]

Dimension of volume = [L3]

The dimension of energy density = [ML2T-2]/[L3] = [ML-1T-2]

So, the dimension of  energy density is  [ML-1T-2]

Meaning of Energy Density

Energy density means the amount of energy stored per unit volume. It indicates the amount of energy of a unit volume of space. We measure energy density in different scenarios. For example, protons and neutrons inside a nucleus are held together by an energy called nuclear binding energy. To obtain an accurate measurement of the energy, we calculate energy density. The concept of energy density is useful in different biological calculations such as energy derived from food and oxygens by animals during chemical reactions.

Energy density plays a vital role in the field of magnetism. With the help of energy density, we deduce different expressions regarding mutual and self-inductance.

Unit of energy density: the SI base unit for energy density is kg/m.s2, but other units such as j/lit w.h/lit are also used. 

Energy density is a scalar quantity as we only talk about the magnitude; no directions are involved.  

Dimensions in Units and Measurements

The dimensional formula of any bodily amount is defined as the expression that represents how and which bottom portions are protected in that amount. It is denoted through enclosing symbols for base portions with the suitable strength in rectangular brackets, i.e., [ ]. An example is the dimension formula of mass, which is given as (M).  Acceleration due to gravity is the acceleration won with the aid of using an item because of gravitational force. Its SI unit is m/s². Each has value and direction. Hence, it is a vector quantity. Acceleration due to gravity is represented with the aid of using g. The general gravity of the earth is given as  9.8m/s².

Examples to write dimensions

Area of a rectangle

Let’s take the formula of the area of the rectangle

Area of the rectangle = length x breadth

= l x l    (where breadth is also showing the length of the side)

= [L1] X [L1]

= [L2]

Here, we can see the length to the power of 2 and cannot find the dimension of mass and time. Hence, the dimension of the area of a rectangle is written as [M0 L2 T0].

Dimensional formula 

The dimensional formula depicts the dependency of physical quantity with fundamental physical quantity, along with the powers.

For example

The formula of speed

Let’s take the formula of speed

Speed = Distance / Time

The distance can be written in length [L]

Time can be written as [T]

The dimensional formula would be [ M0 L1 T-1]

Hence, we can conclude that the speed is dependent on only length and time, not mass.

Dimensional equation

The physical quantity is equated with the dimensional formula to get the dimensional equation. 

Example

Velocity =  [ M0 L1 T-1].

Uses of Dimensional Formula

A dimensional formula can be a helpful tool to check the consistency and coherency of your dimensional equation. The dimensional formula can also be used to establish the correlation between the physical quantities of physical phenomena. These formulas can also be used to change units from one system to another.

Application of Dimensional Analysis

In real-life physics, dimensional analysis is a crucial part of measurement. We use dimensional analysis for the following reasons:

• To ensure that a dimensional equation is consistent;
• Determining the relationship between physical quantities in physical phenomena;
• To switch from one unit system to another.

Limitations of Dimensional Formulas

Dimensional formulas are not concerned with the dimensional constant. Moreover, any formula that contains functions like trigonometric, exponential, logarithmic, and the like cannot be derived. In the context of a physical quantity being a scalar or vector, it does not provide any information about whether the quantity is physical or not.

Conclusion

In this chapter on the dimension formula for energy density, we learnt the basics of energy density and its dimensional analysis. We also learnt its different use cases and the reason behind using energy density. When dealing with an appreciably large system, we cannot determine its exact energy density. To make calculations more accurate, we thus derive energy density.