Stopping current and <PERSON> equation compliance
Preamble: Mathematically, the divergence of an eddy field is zero, thus for the magnetic field $$\nabla\cdot\nabla\times\boldsymbol B = \boldsymbol 0$$ and from the $\nabla\times\boldsymbol B$ Maxwell equation $$\nabla\cdot \left( \boldsymbol J + \varepsilon_0\frac{\partial \boldsymbol E}{\partial t} \right) = 0 \ .$$ The integral of the above over any volume is $0$. So is a closed surface integral (<PERSON> integration theorem) of the divergence's argument, i.e. $\forall A$ $$\oint_A \left( \boldsymbol J + \varepsilon_0\frac{\partial \boldsymbol E}{\partial t} \right) \cdot d\boldsymbol S = 0$$
My thought experiment: As closed surface I choose a sphere $A$ centered at the origin. 903
I split the sphere into a left part $L$ and right part $R$, both open surfaces with $A = L \cup R$. Surfaces $L$ and $R$ have the same orientation such that \begin{equation} \int_L \left( \boldsymbol J + \varepsilon_0\frac{\partial \boldsymbol E}{\partial t} \right) \cdot d\boldsymbol S = \int_R \left( \boldsymbol J + \varepsilon_0\frac{\partial \boldsymbol E}{\partial t} \right) \cdot d\boldsymbol S \end{equation} follows from $$\oint_{L \cup R} \left( \boldsymbol J + \varepsilon_0\frac{\partial \boldsymbol E}{\partial t} \right) \cdot d\boldsymbol S = 0 \ .$$ Now I assume all current $\boldsymbol J$ flowing into the sphere is coming from the $L$-side and then stops inside (in a symmetric pattern around the origin), causing an accumulation of charge and a non-zero $\partial \boldsymbol E / \partial t$.
My problem: Due to the above, $\partial \boldsymbol E / \partial t$ has to be radially symmetric and so \begin{equation} \int_L \frac{\partial \boldsymbol E}{\partial t} \cdot d\boldsymbol S = \int_R \frac{\partial \boldsymbol E}{\partial t} \cdot d\boldsymbol S \end{equation}
EDIT
With my declaration of $L$ and $R$ orientation, this should be \begin{equation} \int_L \frac{\partial \boldsymbol E}{\partial t} \cdot d\boldsymbol S = -\int_R \frac{\partial \boldsymbol E}{\partial t} \cdot d\boldsymbol S \end{equation} which was my error and in hindsight renderes the rest of the question void!
/EDIT
are equal and what remains to fulfill the earlier flux equation is \begin{equation} \int_L \boldsymbol J \cdot d\boldsymbol S = \int_R \boldsymbol J \cdot d\boldsymbol S \ . \end{equation} But clearly we have \begin{align} \int_L \boldsymbol J \cdot d\boldsymbol S & \neq 0 \ \int_R \boldsymbol J \cdot d\boldsymbol S & = 0 \ . \end{align}
So, where is my flaw? What is the missing piece to save the equations? Is this a naive consideration and I need to use the full set of <PERSON> equations and consider an EM wave being emitted by deceleration of charge, thus yielding other sources of $\partial \boldsymbol E / \partial t$? Or include the cause of the stopping current in terms of an electric field? 903