How to Find Energy Stored in a Dielectric Medium

The energy stored in a dielectric medium is a crucial parameter in various electrical and electronic applications, such as capacitors, energy storage devices, and high-voltage insulation systems. To accurately determine the energy stored in a dielectric medium, we can use a well-established formula derived from the principles of electromagnetism.

Understanding the Energy Density in a Dielectric Medium

The energy density in a dielectric medium is given by the formula:

U = 1/2 * ε * E^2 (1)

where:
– U is the energy density (in J/m^3)
– ε is the dielectric constant of the medium (dimensionless)
– E is the electric field strength (in V/m)

This formula is derived from the energy density in an electric field, which is given by:

U = 1/2 * ε₀ * E^2 (2)

where ε₀ is the permittivity of free space (8.854 × 10^-12 F/m).

When a dielectric medium is introduced, the electric field is reduced by a factor of κ, where κ is the dielectric constant. Therefore, the energy density in the dielectric medium is given by:

U = 1/2 * κ * ε₀ * (E/κ)^2 = 1/2 * ε * E^2 (3)

where ε = κ * ε₀ is the permittivity of the dielectric medium.

Calculating the Total Energy Stored in a Dielectric Medium

To measure the total energy stored in a dielectric medium, we can use the formula:

W = ∫U dV (4)

where:
– W is the total energy stored in the medium (in J)
– U is the energy density (in J/m^3)
– dV is the volume element (in m^3)

This formula can be derived from the definition of electric energy within a volume Ω:

W = ∫Ω φ * ρ d^3r (5)

where:
– φ is the electric potential (in V)
– ρ is the charge density (in C/m^3)
– d^3r is the volume element (in m^3)

By using the relation ∇ · D = ρ, we can rewrite the above equation as:

W = ∫Ω ∇ · (φ * D) d^3r – ∫Ω ∇φ · D d^3r (6)

Using the divergence theorem, we can convert the first term on the right-hand side to a surface integral:

∫Ω ∇ · (φ * D) d^3r = ∫∂Ω φ * D · dS (7)

If the dielectric medium is of finite spatial extent, we can neglect the surface term to give:

W = – ∫Ω ∇φ · D d^3r (8)

Using the relation D = ε * E, we can rewrite the above equation as:

W = – ∫Ω ∇φ · (ε * E) d^3r (9)

Using the relation E = – ∇φ, we can rewrite the above equation as:

W = 1/2 * ∫Ω ε * E^2 d^3r (10)

Comparing this equation with equation (1), we can see that the energy density in the dielectric medium is given by:

U = 1/2 * ε * E^2 (11)

Therefore, the total energy stored in the dielectric medium is given by:

W = ∫U dV = ∫(1/2 * ε * E^2) dV (12)

This formula can be used to calculate the energy stored in a dielectric medium, given the electric field strength and the dielectric constant.

Example: Energy Stored in a Parallel-Plate Capacitor

Consider a parallel-plate capacitor with plate area A, plate separation d, and dielectric constant κ. The electric field strength between the plates is given by:

E = σ / κ * ε₀ (13)

where σ is the surface charge density on the plates. The energy stored in the capacitor is given by:

W = 1/2 * C * V^2 = 1/2 * (κ * ε₀ * A / d) * (Q / A)^2 = 1/2 * (κ * ε₀ / d) * Q^2 (14)

where:
– C is the capacitance
– V is the voltage across the plates
– Q is the charge on the plates

Numerical Example

Suppose the dielectric medium has a dielectric constant of κ = 4 and a volume of 1 cm^3, and the electric field strength is 10^6 V/m. The energy density in the dielectric medium can be calculated as:

U = 1/2 * ε * E^2 = 1/2 * (4 * 8.854 x 10^-12 F/m) * (10^6 V/m)^2 = 1.77 x 10^-5 J/m^3 (15)

The total energy stored in the dielectric medium is:

W = U * V = (1.77 x 10^-5 J/m^3) * (1 cm^3) = 1.77 x 10^-7 J (16)

This is the amount of energy stored in the dielectric medium due to the electric field.

In summary, the energy stored in a dielectric medium can be calculated using the formula:

W = ∫(1/2 * ε * E^2) dV (17)

where ε is the dielectric constant, E is the electric field strength, and the integral is taken over the volume of the dielectric medium. This formula can be used to calculate the energy stored in various dielectric media, such as capacitors, dielectric slabs, and dielectric spheres.

References:
– Problem on Energy Density in Dielectric Medium – Physics Forums, 2014-12-21, https://www.physicsforums.com/threads/problem-on-energy-density-in-dielectric-medium.788572/
– 4.4.3 Energy in a Dielectric System – YouTube, 2012-10-05, https://www.youtube.com/watch?v=J4w3vLbY4aI
– Dielectric Constant – an overview | ScienceDirect Topics, https://www.sciencedirect.com/topics/biochemistry-genetics-and-molecular-biology/dielectric-constant
– Energy Stored in a Dielectric Medium – Engineering LibreTexts, 2023-10-03, https://eng.libretexts.org/Bookshelves/Electrical_Engineering/Electro-Optics/Electromagnetic_Field_Theory:_A_Problem_Solving_Approach_%28Zahn%29/03:_Polarization_and_Conduction/3.08:_Energy_Stored_in_a_Dielectric_Medium
– Chapter 4, Lecture 7: Energy Stored in a Dielectric – YouTube, 2022-11-28, https://www.youtube.com/watch?v=GVTmLDGSdnw