The radiation limit depends on the observation-time path

The data in the Example exist: by [F1] one may choose a dimensionless $f\in C_c^{\mathrm{\infty }}(B_a(0))$ with $0\le f\le 1$ and $f=1$ on some ball $B_{r_0}(0)$ with $0<r_0<a$. Then $\int fdy\ge {\lambda }_3(B_{r_0})>0$ by [F3]. For any $f$ admitted in the Example write $F:=\int fdy>0$. Choose $G$ by applying the smooth cutoff supplier to a compact interval slightly larger than $[u-a\mathrm{/}c,u+a\mathrm{/}c]$ and multiply that dimensionless cutoff by $\alpha (t-u{)}^2$. The resulting $G$ is compactly supported, has units $\mathrm{C}{\mathrm{m}}^{-2}$, equals $\alpha (t-u{)}^2$ on a neighborhood of the delay interval, and satisfies $G^{\mathrm{\prime }\mathrm{\prime }}=2\alpha$ there.

The maps $\rho ,J$ in the Example are jointly smooth and compactly supported in $B_a$ in space and in $\mathrm{supp}G\cup \mathrm{supp}{G}^{\mathrm{\prime }}$ in time. Their derivatives obey ${\rho }_t=-G^{\mathrm{\prime }}{\mathrm{\partial }}_{3}f$ and $\mathrm{div}J=G^{\mathrm{\prime }}{\mathrm{\partial }}_{3}f$, so ${\rho }_t+\mathrm{div}J=0$. Thus the retarded-history theorem applies.

For $n=e_1$, every source point in $B_a$ has $\mathrm{\mid }{y}_1\mathrm{\mid }<a$, so $G^{\mathrm{\prime }\mathrm{\prime }}(u+y_1\mathrm{/}c)=2\alpha$. The shifted-current moment of [F2] is therefore ${\displaystyle V(e_1,u)={\int }_{B_a}{J}_t(y,u+y_1\mathrm{/}c)dy=2\alpha F{e}_3\mathrm{.}}$ Since $e_1\cdot e_3=0$, $e_1\times (e_1\times V(e_1,u))=-2\alpha F{e}_3\mathrm{\ne }0$. Apply [F2] on any nondegenerate compact interval containing $u$ to obtain ${\displaystyle E_{\mathrm{r}\mathrm{a}\mathrm{d}}(e_1,u)=-\frac{{\mu }_0\alpha F}{2\pi }{e}_3\mathrm{\ne }0,}$ so $rE(r{e}_1,u+r\mathrm{/}c)$ converges to a nonzero vector.

Let $t_f\in \mathbb{R}$ be a fixed observation time and choose $t_{-}<\inf (\mathrm{supp}G\cup \mathrm{supp}{G}^{\mathrm{\prime }})$. For $y\in B_a$ and $x=rn$, $\mathrm{\mid }x-y\mathrm{\mid }\ge r-a$, so the source retarded time satisfies $t_f-\mathrm{\mid }x-y\mathrm{\mid }\mathrm{/}c\le t_f-(r-a)\mathrm{/}c$. If $r>a+c(t_f-t_{-})$, this is less than $t_{-}$ for every $y\in B_a$, with a uniform strict margin. The source and all its derivatives vanish on a time neighborhood of every such argument; because the inequalities remain strict on a neighborhood of $(x,t_f)$, the retarded potentials are identically zero there and their derivatives vanish. Hence $E(rn,t_f)=B(rn,t_f)=0$ for all such $r$. This fixed-time limit and the nonzero scaled retarded-time limit of step 3.1 are different paths to infinity, not contradictory field values.