= Solution
Use the real <magnetic field> $\boldsymbol{\mathcal B}$ in the coupled equations; its leading periodic approximation will be $\operatorname{Re}(\mathbf B e^{i\omega t})$. In the conducting <incompressible flow>, the <magnetohydrodynamic momentum equation>, <resistive induction equation> and <solenoidal magnetic-field constraint> are
$$
\begin{aligned}
\rho[\partial_t\mathbf u+(\mathbf u\cdot\nabla)\mathbf u]&=-\nabla p+\rho\nu\nabla^2\mathbf u+\mathbf j\times\boldsymbol{\mathcal B},\\
\partial_t\boldsymbol{\mathcal B}&=\nabla\times(\mathbf u\times\boldsymbol{\mathcal B})+\eta\nabla^2\boldsymbol{\mathcal B},\\
\nabla\cdot\mathbf u&=0,\qquad\nabla\cdot\boldsymbol{\mathcal B}=0,\qquad\mathbf j=\mu_0^{-1}\nabla\times\boldsymbol{\mathcal B}.
\end{aligned}
$$
Here $p$ is <pressure>, $\mathbf j$ is <electric current density>, and $\eta$ and $\nu$ are <magnetic diffusivity> and <kinematic viscosity>. The <moving-conductor Ohm law> gives $\mathbf E=-\mathbf u\times\boldsymbol{\mathcal B}+\eta\nabla\times\boldsymbol{\mathcal B}$.
In the insulating exterior, neglecting <displacement current>, the <magnetic field> is <curl>-free away from the wire and <solenoidal>. The imposed <electric current density> supplies
$$
\nabla\times\boldsymbol{\mathcal B}_{\rm ext}=\mu_0J\cos(\omega t)\delta(x)\delta(y+b)\mathbf e_z,
\qquad \nabla\cdot\boldsymbol{\mathcal B}_{\rm ext}=0.
$$
The exterior <electric field> also satisfies <Faraday's law>. At the rigid wall, impose the <no-slip boundary condition> $\mathbf u=0$. At finite conductivity, the <interface conditions for electromagnetic fields> give continuous normal <magnetic field>, continuous tangential <magnetic field> for equal permeability and no prescribed singular surface current, and continuous tangential <electric field>. The normal <electric current density> is zero at an insulating wall; it is automatic for the present two-dimensional fields with current only along $z$. The induced fields decay far from the wire and wall, and initial conditions, or selection of the periodic state after transients, complete the specification. In the limiting <perfect conductor> approximation a surface current can emerge, permitting a tangential-field jump.
The fully coupled solution need not remain monochromatic: a harmonic <magnetic field> produces both a mean and a twice-frequency <Lorentz force density>, and the resulting <velocity> can generate further harmonics. The following parts consistently use the leading magnetic response with fluid motion neglected.
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