= Solution
Work in <cylindrical coordinates> and use the assumed independence of $\varphi$ and $z$. In each material region, the zero <curl> of the static <electric field> gives
$$
\frac{dE_z}{d\rho}=0,\qquad
\frac{d(\rho E_\varphi)}{d\rho}=0.
$$
The tangential <electric field> is zero at a <perfect conductor>. The <dielectric-interface boundary conditions> make its tangential components continuous at each material interface, so $E_z=E_\varphi=0$ throughout the gap. They also vanish in the empty interior and exterior for the regular solution with no applied external <electric field>. Hence
$$
\boxed{\mathbf E=E_\rho(\rho)\mathbf e_\rho.}
$$
This specifies an axis of direction: the sign of $E_\rho$ depends on the sign of $V$.
For the static <magnetic field>, <Gauss's law for magnetism> implies
$$
\frac1\rho\frac{d(\rho B_\rho)}{d\rho}=0.
$$
Thus $\rho B_\rho$ is constant in each layer. Continuity of the normal <magnetic field> across every interface makes that constant global, and regularity at the axis makes it zero. Next use $\nabla\times\mathbf H=0$ away from the conducting surfaces. Its $\varphi$ component gives $dH_z/d\rho=0$. There is no azimuthal <surface current density>, so $H_z$ is continuous across the conducting surfaces and the material interface. The decay of the exterior <magnetic field> at infinity therefore forces this global constant to zero. The only remaining component is
$$
\boxed{\mathbf B=B_\varphi(\rho)\mathbf e_\varphi.}
$$
The <magnetostatic Ampère law> fixes its sign: positive inner <electric current> in the $+z$ direction produces positive $B_\varphi$ for positive <magnetic permeability>.
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