静電エネルギー
場の時間変動がない場合は静電ポテンシャル φ により
U = − ε 0 2 ∫ V E ⋅ grad  φ d 3 x = ε 0 2 ∫ V φ div  E d 3 x − ε 0 2 ∫ V div  ( φ E ) d 3 x = 1 2 ∫ V ρ φ d 3 x − ε 0 2 ∮ ∂ V ( φ E ) ⋅ d S {\displaystyle {\begin{aligned}U&=-{\frac {\epsilon _{0}}{2}}\int _{V}{\boldsymbol {E}}\cdot \operatorname {grad} \phi \,d^{3}x\\&={\frac {\epsilon _{0}}{2}}\int _{V}\phi \,\operatorname {div} {\boldsymbol {E}}\,d^{3}x-{\frac {\epsilon _{0}}{2}}\int _{V}\operatorname {div} (\phi {\boldsymbol {E}})\,d^{3}x\\&={\frac {1}{2}}\int _{V}\rho \,\phi \,d^{3}x-{\frac {\epsilon _{0}}{2}}\oint _{\partial V}(\phi {\boldsymbol {E}})\cdot d{\boldsymbol {S}}\\\end{aligned}}}