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## PowerPedia:Maxwell Equations with Magnetic Charges

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#### Introduction

It is well known that Heaviside was the first to introduce magnetic charges into Maxwell’s electrodynamics. The magnetic charges and currents introduction is justified by the fact that each permanent magnet may be seen as a system of two magnetic charges, which are the magnet’s poles. Accordingly, magnets movement may be considered as magnetic current passage. We may note also that lately monopoles and magnetic current have been discovered [1].

In the book [2] discusses some tasks with the electric and magnetic charges, and in book [3] - the application of these tasks to describe the energy generators with permanent magnets.

#### Maxwell Equations with Magnetic Charges

Let us consider a system of symmetrical Maxwell equations in Cartesian coordinates [2]. Let us denote:

: ~E - electric field strength,

: ~H - magnetic field strength,

: ~\mu - absolute permeability,

: ~\epsilon - absolute dielectric constant,

: ~\vartheta - electric conductivity,

: ~\varsigma - magnetic conductivity,

: ~\varphi - electric scalar potential,

: ~\phi - magnetic scalar potential,

: ~\rho - electric charge density,

: ~\sigma - magnetic charge density.

This system looks as follows:

: \frac{\partial H_z}{\partial y} - \frac{\partial H_y}{\partial z} - \epsilon \frac{\partial E_x}{\partial t} + \vartheta \frac{d \varphi}{dx} =0 ,

: \frac{\partial H_x}{\partial z} - \frac{\partial H_z}{\partial x} - \epsilon \frac{\partial E_y}{\partial t} + \vartheta \frac{d \varphi}{dy} =0 ,

: \frac{\partial H_y}{\partial x} - \frac{\partial H_x}{\partial y} - \epsilon \frac{\partial E_z}{\partial t} + \vartheta \frac{d \varphi}{dz} =0 ,

: \frac{\partial E_z}{\partial y} - \frac{\partial E_y}{\partial z} + \mu \frac{\partial H_x}{\partial t} - \varsigma \frac{d \phi}{dx} =0 ,

: \frac{\partial E_x}{\partial z} - \frac{\partial E_z}{\partial x} + \mu \frac{\partial H_y}{\partial t} - \varsigma \frac{d \phi}{dy} =0 ,

: \frac{\partial E_y}{\partial x} - \frac{\partial E_x}{\partial y} + \mu \frac{\partial H_z}{\partial t} - \varsigma \frac{d \phi}{dz} =0 ,

: \frac{\partial E_x}{\partial x} + \frac{\partial E_y}{\partial y} + \frac{\partial E_z}{\partial z} - \frac{\rho}{\epsilon} =0 ,

: \frac{\partial H_x}{\partial x} + \frac{\partial H_y}{\partial y} + \frac{\partial H_z}{\partial z} - \frac{\sigma}{\mu} =0 .

Let us mark some characteristic features of this systems equations:

the existence of magnetic charges and currents is assumed,

instead of electric and magnetic charges the scalar potentials and conductivities, both electric and magnetic, are introduced.

#### The Problem Statement

We shall consider a system containing magnetic and electric charges, whose density distributions are described by the functions:

: ~\rho(x,y,z,t)= \rho_o \cosh (\beta z + vt) \cosh (\theta y) \delta (x),

: ~\sigma(x,y,z,t)= \sigma_o \cosh (\beta z + vt) \cosh (\theta y) \delta (x),

where

: ~\rho_o, \sigma_o - amplitudes,

: ~\beta, \theta, v - known constants,

: ~\delta (x) - Dirac delta function.

In [3] shows that such functions can describe the charges distribution in magnet motors. In this case the x-axis is directed along the axis of the permanent magnet, and the z-axis is directed along the velocity of the permanent magnet. Dirac delta function describes a layer of charges at the end (x=0) of the permanent magnet, and function cosh describes the charge distribution along the diameter of the permanent magnet end.

#### The Solution of Maxwell Equations

If charges density distribution functions have (if x>0) the form above, then the solution of symmetrical Maxwell is as follows:

: ~E_x(x,y,z,t)=e_x \cosh (\beta z + vt) \sinh (\theta y) \cos(\chi x),

: ~E_y(x,y,z,t)=e_y \cosh (\beta z + vt) \cosh (\theta y) \sin(\chi x),

: ~E_z(x,y,z,t)=e_z \sinh (\beta z + vt) \sinh (\theta y) \sin(\chi x),

: ~H_x(x,y,z,t)=h_x \cosh (\beta z + vt) \cosh (\theta y) \cos(\chi x),

: ~H_y(x,y,z,t)=h_y \cosh (\beta z + vt) \sinh (\theta y) \sin(\chi x),

: ~H_z(x,y,z,t)=h_z \sinh (\beta z + vt) \cosh (\theta y) \sin(\chi x),

: ~\varphi(x,y,z,t)=\varphi_o \sinh (\beta z + vt) \sinh (\theta y) \sin(\chi x),

: ~\phi(x,y,z,t)=\phi_o \sinh (\beta z + vt) \cosh (\theta y) \cos(\sin x),

where

: ~e_x, e_y, e_z, h_x, h_y, h_z, \varphi_o, \phi_o

- coefficients depending on the ~\rho_o, \sigma_o, \beta, \theta, and

: ~\chi=\sqrt {\beta^2+\theta^2}.

For fixed values ??of the variables we have:

: ~E_x(x,t)=e_x^\prime \cosh (vt) \cos(\chi x),

: ~E_y(x,t)=e_y^\prime \cosh (vt) \sin(\chi x),

: ~E_z(x,t)=e_z^\prime \sinh (vt) \sin(\chi x),

: ~H_x(x,t)=h_x^\prime \cosh (vt) \cos(\chi x),

: ~H_y(x,t)=h_y^\prime \cosh (vt) \sin(\chi x),

: ~H_z(x,t)=h_z^\prime \sinh (vt) \sin(\chi x).

It is seen that the vectors satisfy the definition of Energy-dependent Electromagnetic Wave.

#### References

:1. S.T. Bramwell, S.R. Giblin, S. Calder, R. Aldus, D. Prabhakaran, T. Fennell. Measurement of the charge and current of magnetic monopoles in spin ice Nature 461, 956-959 (15 October 2009) | doi: 10.1038/nature08500 Received 18 June 2009 Accepted 14 September 2009.

:2. Khmelnik S.I. Variational Principle of Extremum in electromechanical and electrodynamic Systems. Publisher by “MiC”, printed in USA, Lulu Inc., ID 1142842, Israel, 2010, second edition, ISBN 978-0-557-08231-5, USA, Lulu Inc., ID 1142842

:3. Khmelnik S.I. Energy processes in free-full electromagnetic generators. Publisher by “MiC”, printed in USA, Lulu Inc., ID 10292524, Israel, 2011, second edition, ISBN 978-1-257-08919-2, USA, Lulu Inc., ID 10292524

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