The correlative potential function and a new method for solving Maxwell equations (Q1061941)

This paper is a continuation of the paper ''To construct a vector field with given curl function and divergence function'' by the first author [ibid. 2, 607-612 (1981)]. A new potential function, called the correlative potential function which is different from the classical scalar potential function and vector potential function devoted by Helmholtz has been developed and the new formula of the solution of equations \(\nabla \times \vec f=\vec e_ 1\omega_ 1+\vec e_ 2\omega_ 2+\vec e_ 3\omega_ 3={\vec \omega},\quad \nabla \cdot \vec f=P\) has been given in terms of the correlative potential function \(\psi\), \[ \vec f=-\nabla \psi +\vec e_ 2((1/h_ 2)\int h_ 1h_ 2\omega_ 3du_ 1)+\vec e_ 3(-(1/h_ 3)\int h_ 3h_ 1\omega_ 2du_ 1)-\nabla \psi ' \] where \(\psi\) ' is the solution of the Neumann problem. \(u_ 1\), \(u_ 2\), \(u_ 3\) are orthogonal curvilinear coordinates. \(h_ 1\), \(h_ 2\), \(h_ 3\) are scale factors. Using the correlative potential function, a new method for solving Maxwell equations has been deduced. \[ \Delta \psi e-\mu_ 0\epsilon_ 0(\partial^ 2\psi_ e/\partial t^ 2)=-\rho /\epsilon_ 0-\mu_ 0\int (\partial J_ x/\partial t)dx \] \[ \Delta \psi_ m-\mu_ 0\epsilon_ 0(\partial^ 2\psi_ m/\partial t^ 2)=\int (\partial J_ y/\partial x-\partial J_ x/\partial y)dz \] \[ \vec E=-\nabla \psi e+\vec j\mu_ 0\int (\partial^ 2\psi_ m/\partial z\partial t)dx+ \] \[ \vec K[-\mu_ 0\iint (\partial J_ x/\partial t)dz dx+\mu_ 0\epsilon_ 0\int (\partial^ 2\psi_ e/\partial t^ 2)dz-\mu_ 0\int (\partial^ 2\psi_ m/\partial y\partial t)dx] \] \[ \vec H=- \nabla \psi_ m+\vec i[\int J_ ydz+\mu_ 0\epsilon_ 0\int (\partial^ 2\psi_ m/\partial t^ 2)dx-\epsilon_ 0\int (\partial^ 2\psi_ e/\partial y\partial t)dz]+ \] \[ \vec j[-\int J_ xdz+\epsilon_ 0\int (\partial^ 2\psi_ e/\partial x\partial t)dz] \] where \(\psi_ e\) is the electric type correlative potential. \(\psi_ m\) is the magnetic type correlative potential. The procedure of solving the Maxwell equations are facilitated, we need only to solve two wave equations though the classical method need to solve four wave equations. Several examples for the applications of the correlative potential in electromagnetic theory have been given, the results are identical with those of classical method. Besides, a method for constructing a rotational field with given curl function is proposed.