What is the energy density of an electric field in a dielectric with electric field E and displacement D?

What is the energy density of an electric field in a dielectric with electric field E and displacement D?

• A. u_E = (1/2) E · D
• B. u_E = E^2/(2ε0)
• C. u_E = D^2/(2ε0)
• D. u_E = E · D

Answer: (A)

The energy density in an electric field inside a dielectric is given by one half of the dot product of the electric field and the displacement: u = (1/2) E · D. This form captures how the field stores energy when both E and D are present, not just E alone. Think of building up the field from zero to its final value. The incremental work you do is dW = E · dD. Integrating from zero to the final D gives u = ∫0^D E(D') · dD'. In a linear dielectric, D and E are proportional (D = ε E), so E = D/ε and E · dD = (1/ε) D dD. The integral yields u = (1/2ε) D^2, which is the same as (1/2) E · D since D = ε E. This also reduces to the familiar vacuum result u = (1/2) ε0 E^2 when the medium is vacuum (D = ε0 E). So the correct form, (1/2) E · D, correctly accounts for the material’s response through D, whereas the other expressions either lack the dependence on D, or mix up the ε0 factor, or drop the 1/2.

The energy density in an electric field inside a dielectric is given by one half of the dot product of the electric field and the displacement: u = (1/2) E · D. This form captures how the field stores energy when both E and D are present, not just E alone.

Think of building up the field from zero to its final value. The incremental work you do is dW = E · dD. Integrating from zero to the final D gives u = ∫0^D E(D') · dD'. In a linear dielectric, D and E are proportional (D = ε E), so E = D/ε and E · dD = (1/ε) D dD. The integral yields u = (1/2ε) D^2, which is the same as (1/2) E · D since D = ε E. This also reduces to the familiar vacuum result u = (1/2) ε0 E^2 when the medium is vacuum (D = ε0 E).

So the correct form, (1/2) E · D, correctly accounts for the material’s response through D, whereas the other expressions either lack the dependence on D, or mix up the ε0 factor, or drop the 1/2.