We develop a formalism based on the effective Lagrangian approach (ELA) to evaluate the Chew-Goldberger-Low-Nambu (CGLN) amplitudes for the \(\pi \) -photoproduction process on nucleons, \(\gamma N\rightarrow \pi N'\) . This formalism allows us systematically include any spin \(\frac{1}{2}\) -isospin \(\frac{1}{2}\) , spin \(\frac{1}{2}\) -isospin \(\frac{3}{2}\) , spin \(\frac{3}{2}\) -isospin \(\frac{1}{2}\) , and spin \(\frac{3}{2}\) -isospin \(\frac{3}{2}\) resonance, with both positive and negative parities. In particular, we analyze the effect of the \( P_{33}(1232)\) resonance, the second region four-star \(N^*\) resonances \( P_{11}(1440)\) , \( D_{13}(1520)\) , \( S_{11}(1535)\) , \( P_{33}(1600)\) , and also the third region \(S_{11}(1650) \) and \(P_{11}(1710) \) resonances on total cross-sections and simultaneously on the \(E_{1+}^{3/2}\) and \(M_{1+}^{3/2}\) electromagnetic multipoles. To reproduce the experimental data of the total cross-sections for both proton-induced \(\gamma p \rightarrow n\pi ^+\) and \(\gamma p \rightarrow p\pi ^0\) processes, we describe the strategy for performing the fitting, highlighting the region where each model parameter will be varied, consistently with existing literature information on the same. We have established a reliable set of parameters for the model in accordance with experimental data, which include the coupling constants, the magnetic moments, masses, and widths of the nucleon resonances. With the fitted parameters, we have evaluated the electric and magnetic multipoles, which have shown an excellent agreement with the experimental data, after incorporating the effects of unitarity. Additionally, we have predicted a value for the electric to magnetic ratio of the \(P_{33}(1600)\) resonance, \(R_{EM}=-0.115\) , not yet reported in the literature.
