  Exact 2D analytic expressions for E and B fields and their potentials created
by a linear beam of relativistic charged particles between infinite perfectly
conductive plates and ferromagnetic poles are derived. The solutions are
obtained by summing an infinite sequence of fields from linear charge-images
and current-images in complex space. Knowledge of the normal component of the E
field on the conductor surface makes it possible to calculate the induced
electric charge surface density. In addition, we derive within an improved
linear approximation new analytical expressions for fields near the beam in the
case of an arbitrary beam offset from the median plane. The mathematical
features of exact solutions and limitations for the applicability of linear
approximations are specified.The primary goals of the future high-luminosity
p-p and heavy-ion LHC programme are the search for yet unobserved effects of
physics beyond the SM, searches for rare or low-sensitivity processes in the
Higgs sector, and probing in more detail the mechanism of EW symmetry breaking.
This programme relies on the stable operation of the accelerator. However, as
the beam luminosity increases, a number of destabilizing phenomena occur, in
particular field emission, enhancing the electron cloud effect. For the case of
a proton beam, we apply the exact 2D solution for estimating the intensity of
electron field emission activated by the electric field of the beam in
collimators of the future high-luminosity LHC. Calculation shows that the field
emission intensity is very sensitive to a collimator surface roughness. In
addition, with a relatively small and accidental beam displacement from the
median path, about 20% of the collimator half-gap, the emission intensity
increases by a factor of 1.E+7. This will partially neutralize the beam space
charge, violating acceleration dynamics and enhancing instability effects.
