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There exists a profound difficulty of communication between the people that works in the EMC area in circuit terms and the people that works in field terms. In this paper we show that when the matter is predominantly distributed along a certain direction in space, as for transmission lines, the electromagnetic field can be divided in two modes, each of them with two degrees of freedom, that are practically independent: a longitudinal TM (transverse magnetic) mode and a transversal TE (transverse electric) mode. We also show that two degrees of freedom of the longitudinal mode are the ones that are described by circuit's theory. This formulation is based on the observation that, when the matter is macroscopically described by constitutive laws, the electromagnetic field within the matter can be fully characterized in terms of the potential fields, in total four degrees of freedom. Using the above formulation, we put forward a generalized formulation of the coupling of an external electromagnetic field to a transmission line, valid in any time scale. We apply the above concepts to study, in a common theoretical framework, the iconic case of the conducted and radiated interferences on a transmission line, and we show that: 1-Differently than what is normally assumed in standard transmission-line theory, the normal operation mode and the internally-produced electromagnetic field are predominantly a longitudinal TM mode; 2-The longitudinal mode is affected by both the conducted disturbances and the radiated disturbances; while 3-The transverse mode is affected only by the radiated disturbances. Then, only for systems where the longitudinal mode is predominant, and, the longitudinal and the transversal modes are practically decoupled, EMI can be simulated using circuit simulation software's. Also, to further illustrate the interpretation power of this formulation, we present some other application examples.