How to calculate Energy Density?

The calculation of the amount of energy that may be stored in a given mass of a substance or system is known as energy density. As a result, the higher the energy density of a system or material, the more energy is stored in its mass. Energy may be stored in a wide range of materials and methods. In four different types of reactions, any material can release energy. Nuclear, chemical, electrochemical, and electrical are the four types of energy. In most cases, only useable or extractable energy is assessed when estimating the quantity of energy in a system. 

Energy density 

Energy density is defined as the total amount of energy in a system per unit volume. For example, the number of calories per gram of food. Low-energy-density foods provide fewer calories per gram than high-energy-density foods, allowing you to consume more of them. It is symbolized by the letter U. Magnetic and electric fields can store energy.

Formula for energy density 

• The energy density of a capacitor or electric field is represented as,

Electrical energy density =  permittivity × (Electric field)² /2

U_E = (1/2)ε₀E²

Derivation of Electric field energy density 

Energy density = Energy/volume

U_E = U/V

Energy (U) = 1/2 (ε₀ × E²) × Ad

Volume(V) = Ad

U_E = 1/2 (ε₀ × E²) × Ad/Ad

U_E = (1/2)ε₀E²

• For any inductor or magnetic field, electric density is represented as

Magnetic energy density = magnetic permeability × (magnetic field)²/2

U_B = (1/2μ₀)B²

Derivation of Magnetic field energy density 

Energy density = Energy/volume

U_B = 1/2 (LI²⁾/Al

Flux = NBA = LI

B = μ₀ NI/length

I = B (Length)/ Nμ₀

U_B = (1/2) (B (Length)/ Nμ₀) (NBA)/A (length)

U_B = (1/2μ₀)B²

In the energy density of electromagnetic waves, both electric and magnetic fields contribute. Therefore total energy density is equal to the sum of the electric and magnetic fields. 

U = U_E + U_B

U = (1/2)ε₀E² + (1/2μ₀)B²

Here, in the above-mentioned derivations the parameters are,

U_E = Electrical Energy Density

U_B = Magnetic Energy Density

ε₀ = Permittivity,

μ₀ = Permeability

E = Electric Field

B = Magnetic Field.

Sample Problems

Question 1: What is the energy density?

Answer:

Energy density is defined as the total amount of energy in a system per unit volume.

For the total energy density, the formula is given by U = (1/2)ε₀E² + (1/2)μ₀B²

Question 2: What is the Formula for the energy density of an electric field or a capacitor?

Answer:

The energy density of an electric field or a capacitor is given by

U_B = (1/2μ₀)B²

Where,

U_B = Magnetic Energy Density,
μ₀ = Permeability
B = Magnetic Field.

Question 3: Calculate the energy density of a capacitor with an electric field of E = 15 V/m.

Solution:

Given: E = 15 V/m, ε₀ = 8.8541 × 10^-12 F/m

U_E = (1/2)ε₀E²

U_E = (1/2) × 8.8541 × 10^-12 × 225

U_E = (1/2) × 1992.2× 10^-12

U_E = 996.1 × 10^-12

U_E = 9.96 × 10^-10 FV²/m³

Question 4: Calculate the energy density of a capacitor with an electric field of E = 30 V/m.

Solution:

Given: E = 30 V/m, ε₀ = 8.8541 × 10^-12 F/m

U_E = (1/2)ε₀E²

U_E = (1/2) × 8.8541 × 10^-12 × (30)²

U_E = (1/2) × 7968.69 × 10^-12

U_E = 3.984 × 10^-9 FV²/m³

Question 5: If an inductor with a magnetic field of B = 6 T then calculate its energy density of it.

Solution:

Given: B = 6 T, μ₀ = 4π × 10^-7 NA^-2

U_B = (1/2μ₀)B²

U_B = (1/2 ×4π × 10^-7)× 6²

U_B = (1/25.12 × 10^-7) × 36

U_B = 1.43 × 10^-7 J/m³

Question 6: Calculate the energy density of both electric and magnetic fields, the magnetic field has a value of 2  × 10^-2 T and the electric field has a value of 4 × 10⁷ V/m in a certain space of a region.

Solution:

Given: B = 2 × 10^-2 T

E = 4 × 10⁷ V/m

 μ₀ = 4π × 10^-7 NA^-2

ε₀ = 8.8541 × 10^-12 F/m

Electrical energy density-

U_E = (1/2) ε₀E²

U_E = 1/2 × 8.8541 × 10^-12 × (4 × 10⁷)²

U_E = 7083.28 J/m³.

Magnetic energy density,

U_B = (1/2 × μ₀)B²

U_B = (1/2 × 4π × 10^-7) × (2 × 10^-2)²

U_B = 159.2 J/m³

Total energy density – U = U_E + U_B

U = 7083.28 + 159.2

U = 7242.48 J/m³.