The planar limit is a combinatorial generating function which counts the number of planar graphs of arbitrary valency and this is very closely connected to random matrix calculations. On the other hand an analytic treatment of the random matrix reduces the problem to studying the minimizer of the so called logarithmic energy with external field. We will show an elementary way of dealing with the minimizer of the logarithmic energy with external fields. This is based on manipulations of Chebyshev polynomials and combinatorial identities which give a nice new formula for the minimum of the energy. This indicates why the analyticity claim of the planar limit holds true.