Integro-differential equations for a quasi-stationary electromagnetic field in a nonmagnetic conductive body with a defect

The initial-boundary value problem (IBVP) for the system of Maxwell’s equations for a quasi-stationary approximation is considered in relation to a non-ferromagnetic conductive body in the field of an external current. It is assumed, that the body is not homogeneous in its conductive properties and includes a volume defect in the form of a cavity (a non-conductive subdomain). The IBVP is considered in the classical formulation: the tensions of the electric and magnetic fields are supposed to be continuously derivatively outside the boundary between conductive and non-conductive domains, and continuous at the boundaries of these domains; in this case, the boundaries of the domains are Lyapunov surfaces. On these surfaces, the usual boundary conditions for the tensions of electric and magnetic field must be satisfied: their tangential components are continuous. In addition, tensions decrease quickly at infinity. Based on these assumptions, integro-differential equations for the tensions of electric and magnetic field are derived. These equations take into account the inhomogeneity of the conductor and the presence of the internal defect. The equivalence of the integro-differential equations and the IBVP for the system of Maxwell’s equations is proved for the electromagnetic field inside and outside the conductor.