A microscopic study of elastic scattering of carbon isotopes from different target nuclei at various incident energies is presented by using density distributions derived for $^{10–14,16}\rm{C}$ nuclei using relativistic mean field (RMF) theory. To obtain the real part of the optical potential, the double folding model is used.Woods–Saxon potential is used for the imaginary part. The theoretical results are discussed and compared with each other as well as with the experimental data. It is seen that the agreement between theoretical results and experimental data is very good. Also, new global equations for the imaginary potentials of the $^{10–14,16}\rm{C}$ nuclei are derived from the results of the theoretical analysis.