Energy Density Formula

The quantity of energy that may be stored in a given mass of a substance or system is determined by its energy density. As a result, the higher a system’s or material’s energy density, the more energy it can store in its mass. Many different substances and techniques can be used to store energy. Most typically, only usable or extractable energy is measured when determining the amount of energy in a system. We commonly compute energy density in scientific equations. The energy density formula will be discussed with examples in this article.

Energy Density

The total quantity of energy in a system per unit volume is known as Energy density. The number of calories per gram of food, for instance. Low-energy-density foods have fewer calories per gram, allowing you to eat more of them. It is denoted by U. Energy can be stored in magnetic and electric fields.

Formula of Energy Density

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

U_E = (1/2)ε₀E²

Where,

• U_E = Electrical Energy Density,
• ε₀ = Permittivity,
• E = Electric Field.

Derivation

Energy density = Energy/volume

U_E = U / V

Energy = 1/2 [ε₀ E²] × Ad

U_E = 1/2 [ε₀ E²] × Ad / Ad

U_E = (1/2)ε₀E²

• The energy density of a magnetic field or an inductor is given by,

U_B = (1/2μ₀)B²

Where,

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

Derivation

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

Magnetic and electric fields both contribute to the energy density of electromagnetic waves. As a result, the total energy density of electric and magnetic fields equals the total energy density.

U = (1/2)ε₀E² + (1/2μ₀)B²

Sample Questions 

Question 1: Define Energy density.

Answer:

The total quantity of energy in a system per unit volume is known as energy density. Total energy density involves both capacitive energy density and inductive energy density. The formula for the total energy density,

U = (1/2)ε₀E² + (1/2μ₀)B²

Question 2: What is the Formula of the energy density of a magnetic field or an inductor?

Answer:

The energy density of a magnetic field or an inductor 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 = 12 V/m.

Solution:

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

Since,

U_E = (1/2)ε₀E²

∴ U_E = (1/2) × 8.8541 × 10^-12 × 12²

∴ U_E = (1/2) × 1274.99 × 10^-12

∴ U_E = 637.495 × 10^-12

∴ U_E = 6.375 × 10^-10 FV²/m³

Question 4: In one area of space, the magnetic field has a value of 3 × 10^-2 T. In contrast, the electric field has a value of 3 × 10^-7 V/m. Calculate both the electric and magnetic fields’ total energy density.

Solution:

Given: B = 3 × 10^-2 T, E = 3 × 10^-7 V/m, ε₀ = 8.8541 × 10^-12 F/m, μ₀ = 4π × 10^-7 NA^-2

Since,

U = (1/2)ε₀E² + (1/2μ₀)B²

∴ U = ((1/2) × 8.8541 × 10^-12 × (3 × 10^-7)²) + ((1/(2 × 4π × 10^-7)) × (3 × 10^-2)²)

∴ U = 3982.5 + 358.1

∴ U = 4340.6 J/m³

Question 5: Calculate the energy density of a capacitor with an electric field of E = 20 V/m.

Solution:

Given : E = 20 V/m, ε₀ = 8.8541 × 10^-12 F/m

Since,

U_E = (1/2)ε₀E²

∴ U_E = (1/2) × 8.8541 × 10^-12 × 20²

∴ U_E = (1/2) × 3541.64 × 10^-12

∴ U_E = 1.7 × 10^-9 FV²/m³

Question 6: Calculate the energy density of an inductor with a magnetic field of B = 8 T.

Solution:

Given: B = 8 T, μ₀ = 4π × 10^-7 NA^-2

Since,

U_B = (1/2μ₀)B²

∴ U_B = (1/(2 × 4π × 10^-7)) × 8²

∴ U_B = (1/25.12 × 10^-7) × 64

∴ U_B = 2.5477 × 10^-7 J/m³