  The spoke instability in an $E\times B$ Penning discharge is shown to be
strongly affected by the boundary that is perpendicular to $B$ field lines. The
instability is the strongest when bounded by dielectric walls. With a
conducting wall, biased to collect electron current from the plasma, the spoke
becomes faster, less coherent and localised closer to the axis. The
corresponding anomalous cross-field transport is assessed via simultaneous
time-resolved measurements of plasma potential and density. This shows a
dominant large-scale $E\times B$ anomalous character of the electron
cross-field current for dielectric walls reaching $40-100$% of the discharge
current, with an effective Hall parameter $\beta_\mathrm{eff}\sim10$. The
anomalous current is greatly reduced with the conducting boundary
(characterised by $\beta_\mathrm{eff}\sim10^2$). These experimental
measurements are shown to be qualitatively consistent with the decrease of the
$E$ field that triggers the collisionless Simon-Hoh instability.
