  In this dissertation, we analytically study the apex field enhancement factor
(FEF), $\gamma_a$, by constructing a method which consists in minimizing an
error function defined as to measure the deviation of the potential at the
boundary, yielding approximate axial multipole coefficients of a general
axial-symmetric conducting emitter shape, on which the apex FEF can be
calculated by summing its respective Legendre series. Such method is
analytically applied for a conducting hemisphere on a flat plate, confirming
the known result of $\gamma_a = 3$. Also, it is applied on a hemi-ellipsoid on
a plate where the values of the apex FEF are compared with the ones extracted
from the analytical expression. Then, the method is applied for the hemisphere
on a cylindrical post (HCP) model. In this case, to analytically estimate the
apex FEF from first principles is a problem of considerable complexity. Despite
the slow convergence of the apex FEF, useful analytical conclusions are drawn
and explored, such as, it is confirmed that all even multipole contributions of
the HCP model are zero, which in turn leads to restrictions on the charge
density distribution: it will be shown the surface charge density must be an
odd function with respect to height in an equivalent system. Also, expressions
found for the apex FEF depend explicitly on the aspect ratio, that is, the
ratio of height by base radius. Using the dominant multipole contribution, the
dipole, at sufficient large distances, it is shown that, for two interacting
emitters, as their separation distance increases, the fractional change in apex
FEF, $\delta$, decreases following a power law with exponent $-3$. The result
is extended for conducting emitters having an arbitrary axially-symmetric
shape, where it is also shown $\delta$ has a pre-factor depending on geometry,
confirming the tendency observed in recent analytical and numerical results.
