= Solution
Orient the <unit normal> $\mathbf n$ from $R_1$ to $R_2$ and write $\mathbf H$ for the <magnetic field strength>. For static <electric fields>, $\mathbf E=-\nabla\phi$; the static <Ampère's law> gives $\nabla\times\mathbf H=\mathbf J_f$. The <curl> product rule therefore gives the local <Poynting flux and terminal power> identity
$$
\boldsymbol{\mathcal P}=-\nabla\phi\times\mathbf H
=\phi\mathbf J_f-\nabla\times(\phi\mathbf H).
$$
Integrate over an oriented piece $\Sigma_R$ of the separating surface and apply <Stokes theorem>:
$$
\int_{\Sigma_R}\boldsymbol{\mathcal P}\cdot\mathbf n\,dS
=\int_{\Sigma_R}\phi\mathbf J_f\cdot\mathbf n\,dS
-\oint_{\partial\Sigma_R}\phi\mathbf H\cdot d\boldsymbol\ell.
$$
Once all the wires are included, the first <surface integral> is $\sum_k\phi_kI_k$: the <electric potential> is constant on each <perfect conductor> and the integral of its normal <current density> is its signed <electric current>. Equivalently, remove the wire cross-sections and apply <Stokes theorem> in the surrounding insulator. Each resulting hole has negative boundary orientation, and the <magnetostatic Ampère law> around it gives the positive contribution $\phi_kI_k$.
For a closed separating surface the boundary term is absent. For a noncompact separating surface, take $R\to\infty$ with the usual isolated-circuit condition $\oint_{\partial\Sigma_R}\phi\mathbf H\cdot d\boldsymbol\ell\to0$. In either case,
$$
\boxed{\int_\Sigma\boldsymbol{\mathcal P}\cdot\mathbf n\,dS=\sum_{k=1}^N\phi_kI_k.}
$$
For a steady closed circuit the signed <electric currents> sum to zero, so shifting the reference of <electric potential> does not change the terminal power.
For a fully general noncompact surface, the hypotheses as printed need an additional condition at infinity. The exact result before imposing it is
$$
\boxed{P_{1\to2}=\sum_{k=1}^N\phi_kI_k-
\lim_{R\to\infty}\oint_{\partial\Sigma_R}\phi\mathbf H\cdot d\boldsymbol\ell.}
$$
Time independence alone does not remove this boundary term or ensure a finite <Poynting flux>; for example, spatially constant crossed <electric fields> and <magnetic fields> carry a constant <Poynting vector> through an infinite plane. The stated terminal-power formula holds with the boundary condition specified above.
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