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Energy Density Formula

Energy density is the computation of the amount of energy that can be stored in a given mass of a substance or a system. So, the more the energy density of a system or material, the greater will be the amount of energy stored in its mass. Energy can be stored in many varieties of substances and systems. Any material can release the energy in four types of reactions. Which are nuclear, chemical, electrochemical and electrical. While calculating the amount of energy in a system most often only useful or extractable energy is measured. In scientific equations, we often compute energy density. In this topic, we will discuss the energy density formula with examples. Let us learn it!

Energy Density Formula

Source:en.wikipedia.org

Energy Density Formula

What is energy density?

Energy Density can be defined as the total amount of energy in a system per unit volume. For example, the number of calories available per gram weight of the food. Foodstuffs have low energy density and will provide less energy per gram of the food. It means that we can eat more of them due to fewer calories.

Therefore we may say that energy density is the amount of energy accumulated in a system per unit volume. It is denoted by letter U. Magnetic and electric fields are also the main sources for storing the energy.

Energy Density Formula

In the case of electric field or capacitor, the energy density formula is expressed as below:

Electrical energy density =  \(\frac {permittivity \times Electric field squared} {2}\)In the form of equation,

\(U_E\) = \(\frac{1}{2} \varepsilon_0 E^2\)

The energy density formula in case of magnetic field or inductor is as below:

Magnetic energy density = \(\frac {magnetic field squared} { 2 \times magnetic permeability }\)

In the form of an equation,

\(U_B\) = \(\frac {1}{2 \mu_0} B^2\)

The general energy is:

U = \(U_E + U_B\)

Where,

U Energy density
\(U_E\) Electrical energy density
\(U_B\) Magnetic energy density
\(\varepsilon_0\) Permittivity
E Electric field
B Magnetic field
\(\mu\) magnetic permeability

 

Regarding the electromagnetic waves, both magnetic and electric fields are involved in contributing to energy density equally. Thus, the formula of energy density will be the sum of the energy density of electric and magnetic fields both together.

Solved Examples

Q.1: In a certain region of space, the magnetic field has a value of \(3\times 10^{-2}\) T. And the electric field has a value of \(9 \times 10 ^7 V m^{-1}\). Determine the combined energy density of the electric and magnetic fields both.

Solution: First we have to calculate the density and energy of each field separately. Then we will add the densities to obtain the total energy density.

Given parameters in the question are:

B = \(3\times 10^{-2} T\)

E = \(9 \times 10 ^7 V m^{-1}\)

\(\varepsilon_0 = 8.85 \times 10^{-12} C^2 N^{-1} m^{-2}\)

\(\mu = 4 \times \pi \times 10^{-7} N A^{-2}\) .

Thus electrical energy density,

\(U_E\) = \(\frac{1}{2} \varepsilon_0 E^2\)

= \(\frac{1}{2} \times 8.85 \times 10^{-12} \times (9 \times 10 ^7 )^2 \)

= \(\frac{1}{2} \times 8.85 \times 10^{-12} \times (9 \times 10 ^7 )^2 \)

= 35842.5\( \;J m^{-3}\)

Now, magnetic energy density,

\(U_B\) = \(\frac {1}{2 \mu_0} B^2 \)

= \(\frac {1}{2 \times 4 \times \pi \times 10^{-7} } \times (3\times 10^{-2}) ^2 \)

= \(\frac {1}{2 \times 4 \times 3.14 \times 10^{-7} } \times (3\times 10^{-2}) ^2 \)

= 358.1\(\; J m^{-3}\)

Thus total energy density will be,

U =\( U_E + U_B\)

= 35,842.5 + 358.1

= 36200.6 \(\; J m^{-3}\)

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