What is the electric energy density u in a vacuum or dielectric?

What is the electric energy density u in a vacuum or dielectric?

• A. u = (1/2) ε E^2
• B. u = ε E^2
• C. u = (1/2) E^2/ε
• D. u = (1/2) D·E

Answer: (A)

Electric energy density is the amount of electric energy stored per unit volume in the field. In a vacuum or a linear dielectric, this energy scales with the square of the electric field, and the strength of the medium is captured by its permittivity. Integrating the energy needed to build up the field from zero to its final value yields the familiar result u = (1/2) ε E^2. Here ε is the medium’s permittivity (ε0 in vacuum, ε0εr in a dielectric). You can also write the energy density as u = (1/2) E · D. In linear media, where D = εE, this is exactly the same as (1/2) ε E^2, but in nonlinear or more general media the E · D form remains the correct expression. The other forms either miss the 1/2 factor or rearrange the variables in a way that doesn’t match the energy per volume in these media.

Electric energy density is the amount of electric energy stored per unit volume in the field. In a vacuum or a linear dielectric, this energy scales with the square of the electric field, and the strength of the medium is captured by its permittivity. Integrating the energy needed to build up the field from zero to its final value yields the familiar result u = (1/2) ε E^2. Here ε is the medium’s permittivity (ε0 in vacuum, ε0εr in a dielectric).

You can also write the energy density as u = (1/2) E · D. In linear media, where D = εE, this is exactly the same as (1/2) ε E^2, but in nonlinear or more general media the E · D form remains the correct expression. The other forms either miss the 1/2 factor or rearrange the variables in a way that doesn’t match the energy per volume in these media.