How can you compute the energy stored in a capacitor using the electric field?

How can you compute the energy stored in a capacitor using the electric field?

• A. U = (1/2) ∫ ε0 E^2 dV
• B. U = (1/2) ∫ ε E^2 dV
• C. U = ∫ ε E dV
• D. U = ∫ φ dq

Answer: (B)

The energy stored in an electric field is described by the energy density u = 1/2 ε E^2, where ε is the permittivity of the medium and E is the electric field. To get the total energy, you integrate this density over all space: U = (1/2) ∫ ε E^2 dV. This is the general form for a linear dielectric; in vacuum ε = ε0, giving U = (1/2) ∫ ε0 E^2 dV. If you prefer a capacitor-based view, you can also express the energy as U = 1/2 C V^2, which is equivalent to the field integral when you relate C and V to the geometry and E. The key idea is that energy resides in the field itself, quantified locally by 1/2 ε E^2.

The energy stored in an electric field is described by the energy density u = 1/2 ε E^2, where ε is the permittivity of the medium and E is the electric field. To get the total energy, you integrate this density over all space: U = (1/2) ∫ ε E^2 dV. This is the general form for a linear dielectric; in vacuum ε = ε0, giving U = (1/2) ∫ ε0 E^2 dV. If you prefer a capacitor-based view, you can also express the energy as U = 1/2 C V^2, which is equivalent to the field integral when you relate C and V to the geometry and E. The key idea is that energy resides in the field itself, quantified locally by 1/2 ε E^2.