[28] Consider a scattering object made of a reciprocal material. In that case the equation (24) holds for all times *t*, for all electromagnetic fields {*E*^{a}, *B*^{a}} and {*E*^{b}, *B*^{b}}, and for every closed surface *S*_{r} ⊂ *V* around the point *r*.

[29] Let the two fields *E*^{a}(*r*, *t*), *H*^{a}(*r*, *t*) and *E*^{b}(*r*, *t*), *H*^{b}(*r*, *t*) be the total electric and magnetic fields from the two incident fields

Here *E*_{0}(*t*) is zero except for a finite time interval that for simplicity is chosen to be 0 < *t* < *t*_{1} and the origin is located inside the scattering object. The scattered electric and magnetic fields are in the far zone given by

Since the scattered field has finite energy the farfield amplitudes have to go to zero when *t* − *r*/*c*_{0} gets large. Thus *F*^{a}(, *t*) and *F*^{b}(, *t*) are zero for *t* < 0, due to causality, and negligible for *t* > *T*, for some *T*. Consider the surface *S* to be a sphere, denoted *S*_{R}, with radius *R* and with center at the origin. The radius *R* is large enough so that the surface of the sphere is in the far zone. The identity (24) implies

where

It is first proven that *I*_{1} = *I*_{3} = 0. Do the substitution θ′ = π − θ and ϕ′ = 2π − ϕ in the surface integral of *I*_{1}. After some manipulations it follows that

and hence *I*_{1} = 0. To see that *I*_{3} = 0 one observes that on the surface *S*_{R}

where ⊙ denotes the convolution of a scalar product (cf. equation (25)). It follows that *I*_{3} = 0. The remaining integral *I*_{2} is given by

where

Since *F*(, *t*) is approximately zero for all times except 0 < *t* < *T*, the integrals reduce to

Now *E*_{0}(*t*) is zero except when 0 < *t* < *t*_{1}. Consider the time interval 0 < *t* < *T* and choose the radius of the sphere large enough to satisfy *R* ≫ *c*_{0}*T*. In that case the integrand in *K*_{1}(, *z*, *t*) is nonzero only for and −*R* < *z* < −*R* + *c*_{0}*T* and the integrand in *K*_{2}(, *z*, *t*) is nonzero only for and *R* − *c*_{0}*T* < *z* < *R*. After a substitution θ′ = π − θ in the part of *I*_{2}(*t*) that contains *K*_{1}(, *z*, *t*) it follows that for sufficiently small *T*/*R* the integral *I*_{2}(*t*) is given by

Since *I*_{1}(*t*) = *I*_{3}(*t*) = 0 it follows that *I*_{2}(*t*) = 0 for all times, and in particular for 0 < *t* < *T*. From equation (44) it follows that this can only be fulfilled if

for all times *t*.

[30] In addition to the result in equation (45) there is a reciprocity result also for the case when the incident fields are given by

In that case it is straightforward to see that *I*_{1} = *I*_{3} = 0 and that

Since *I*_{2}(*t*) = 0 for all times it follows that for all times