Exact two-dimensional (2D) analytic expressions for electric and magnetic 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 electric 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 Large Hadron Collider programme after the Long Shutdown 2 are the search for yet unobserved effects of physics beyond the Standard Model, searches for rare or low-sensitivity processes in the Higgs sector, and probing in more detail the mechanism of electroweak 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 Large Hadron Collider. 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 107. This will partially neutralize the beam space charge, violating acceleration dynamics and enhancing instability effects.
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