Energy and Poyntings Theorem

The Lorentz force in Eq. (2.104) acting on the N particles will change the total energy Emat of the matter inside the system by 

That is, electromagnetic energy of the field can be converted into mechanical energy of the particles and vice versa. Rewriting this expression gives 

The important identity in Eq. (2.107) relates the energy density of the electromagnetic field with the currents flowing in the system and is sometimes denoted as Foynting s theorem. Inserting these results in Eq. (2.106) and taking the arbitrariness of the volume V into account, we find 

Note that the total time derivative acting on the integral in Eq. (2.106) had to be replaced by a partial time derivative since the order of integration and differentiation has been interchanged. Subsequent spatial integration over the volume V and application of Gauss theorem (cf. appendix A.1.3) yields 

For sufficiently large volumes V the electromagnetic fields are completely embedded in this volume V and no fields are leaving this volume. Hence the right-hand side of Eq. (2.111) vanishes and 

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