  We analyse linear stability of interfacial waves in an idealised model of an
aluminium reduction cell consisting of two stably stratified liquid layers
which carry a vertical electric current in a collinear external magnetic field.
If the product of electric current and magnetic field exceeds a certain
critical threshold depending on the cell design, the electromagnetic coupling
of gravity wave modes can give rise to a self-amplifying rotating interfacial
wave which is known as the metal pad instability. Using the eigenvalue
perturbation method, we show that, in the inviscid limit, rectangular cells of
horizontal aspect ratios $\alpha=\sqrt{m/n}$, where $m$ and $n$ are any two odd
numbers, can be destabilised by an infinitesimally weak electromagnetic
interaction while cells of other aspect ratios have finite instability
thresholds. This fractal distribution of critical aspect ratios, which form an
absolutely discontinuous dense set of points interspersed with aspect ratios
with non-zero stability thresholds, is confirmed by accurate numerical solution
of the linear stability problem. Although the fractality vanishes when viscous
friction is taken into account, the instability threshold is smoothed out
gradually and its principal structure, which is dominated by the major critical
aspect ratios corresponding to moderate values of $m$ and $n$, is
well-preserved up to relatively large dimensionless viscous friction
coefficients $\gamma\sim 0.1$. With a small viscous friction, the most stable
are cells with $\alpha^{2}\approx2.13$ which have the highest stability
threshold corresponding to the electromagnetic interaction parameter
$\beta\approx 4.7$.
