電荷保存則
電荷 qi が時間的に変化しないとすれば
∂ ρ ∂ t = ∑ i q i ∂ ∂ t δ 3 ( r − r i ( t ) ) {\displaystyle {\frac {\partial \rho }{\partial t}}=\sum _{i}q_{i}{\frac {\partial }{\partial t}}\delta ^{3}({\boldsymbol {r}}-{\boldsymbol {r}}_{i}(t))}
= − ∑ i q i v i ( t ) ⋅ ∇ δ 3 ( r − r i ( t ) ) {\displaystyle =-\sum _{i}q_{i}{\boldsymbol {v}}_{i}(t)\cdot \nabla \delta ^{3}({\boldsymbol {r}}-{\boldsymbol {r}}_{i}(t))}
= − ∇ ⋅ j ( r , t ) {\displaystyle =-\nabla \cdot {\boldsymbol {j}}({\boldsymbol {r}},t)}
従って、
∂ ρ ( r , t ) ∂ t + ∇ ⋅ j ( r , t ) = 0 {\displaystyle {\frac {\partial \rho ({\boldsymbol {r}},t)}{\partial t}}+\nabla \cdot {\boldsymbol {j}}({\boldsymbol {r}},t)=0}
が成立つ。