= Solution
In vacuum, the source-free <Maxwell equations> are
$$
\nabla\cdot\mathbf E=\nabla\cdot\mathbf B=0,\qquad
\nabla\times\mathbf E=-\partial_t\mathbf B,\qquad
\nabla\times\mathbf B=c^{-2}\partial_t\mathbf E.
$$
Put $\psi=\omega(t-z/c)$ and denote the constant electric amplitude by $E_0$. A corresponding traveling-wave <magnetic field> is
$$
\boxed{\mathbf B=\frac{E_0}{c}(-\sin\psi,\cos\psi,0).}
$$
Both <divergences> vanish because both the <electric field> and the <magnetic field> depend only on $z,t$ and have zero $z$ component. Direct differentiation gives $\nabla\times\mathbf E=(\omega E_0/c)(\cos\psi,\sin\psi,0)=-\partial_t\mathbf B$ and $\nabla\times\mathbf B=(\omega E_0/c^2)(-\sin\psi,\cos\psi,0)=c^{-2}\partial_t\mathbf E$. A time-independent background <magnetic field> satisfying the homogeneous equations could be added; the formula is the wave component.
Introduce the <magnetic vector potential> and <electric potential> by $\mathbf B=\nabla\times\mathbf A$ and $\mathbf E=-\nabla\phi-\partial_t\mathbf A$. These identities automatically give the divergence equation for $\mathbf B$ and <Faraday's law>. For this wave choose $\phi=0$ and $\mathbf A=(c/\omega)\mathbf B$; the <derivatives> just calculated verify both potential equations.
The electromagnetic <force> is the <Lorentz force>, $Q(\mathbf E+\mathbf v\times\mathbf B)$. On the smooth constraint plane $z=z_0$, the <constraint force> is normal to the plane. Both $\mathbf v$ and $\mathbf B$ are tangential, so $\mathbf v\times\mathbf B$ is normal and is balanced by that <constraint force>; it does not affect the tangential motion. Thus
$$
\ddot{\mathbf r}_\parallel=\frac{QE_0}{m}(\cos\psi,\sin\psi).
$$
A particular motion is $\mathbf r_\parallel=\mathbf r_0-[QE_0/(m\omega^2)](\cos\psi,\sin\psi)$, a circle of radius
$$
\boxed{\frac{|QE_0|}{m\omega^2}.}
$$
The general planar motion also includes a constant drift <velocity>; the particle moves on a circle when that drift is zero. The <constraint force> can do no work because it is perpendicular to the <velocity>.
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