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The concept of the generalized scattering amplitude is applied to the electromagnetic (EM) scattering of an arbitrarily shaped, perfectly conducting object partially immersed in a semi-infinite dielectric medium. The dielectric medium can have either electric loss or magnetic loss or both. In a two-dimensional (2-D) formulation, after the outgoing cylindrical wave is factored out from the scattered wave, the remaining wave envelope component in the scattered wave is defined as the generalized scattering amplitude. The transformed Helmholtz equation in terms of the generalized scattering amplitude can be solved numerically using a finite-difference method over the entire scattering domain including both the semi-infinite free-space (vacuum or air) and the semi-infinite dielectric medium. Example problems of scattering by infinitely long, perfectly conducting cylinders of circular and trapezoidal ship-shaped cross sections are solved to demonstrate the theoretical formulation and numerical method. The radial profiles of the generalized scattering amplitude and the total field over the entire scattering region are also presented, and their properties are discussed. The far-field bistatic cross section and induced current density on the obstacle's surface are also presented. These results show that the method can be used to yield complete and accurate solutions to 2-D EM scattering problems involving arbitrarily shaped metallic objects partially immersed in a penetrable semi-infinite dielectric medium.