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The analytical expressions of electric fields inside and outside a magnetized cold plasma sphere are presented by reforming the spherical electromagnetic parameter based on the scales transformation of electromagnetic theory. The obtained results are in good agreement with that in literatures. The angle between the direction of inside field and that of outside field is derived. In S wave band, numerical calculations of effects induced on the inner field by parameters are established. Simulations show that the angle between incident field and the outside magnetic field influences the inner field remarkably. The inner field will decrease as the electron density increasing, however, this density has a great affect on the inner field’s direction. The magnitude of the inner field is proportional to the incident wave’s frequency.

The investigations both for electromagnetic (EM) scattering features and their applications of spherical target have been of a great interest. The electric fields inside and outside a single isotropic dielectric sphere have been researched [1-4]. The scattering features of an isotropic dielectric sphere and a conducting sphere, which are irradiated by an EM wave propagating in the z-direction and polarizing in the x-direction, have been studied [5,6]. By using method of rotating coordinate system, the scattering properties of spherical targets that are irradiated by a wave from an arbitrary direction is studied [

Many particles are practically anisotropic and much smaller than the wave length in size, such as raindrops, sand-dust storm particulates, fog droplets, etc. Therefore the problems relative to electromagnetic field may be approximately considered as an electrostatic one [

Assume a magnetized cold plasma sphere to have radius R_{0} and its centre to be located at the origin of the primary coordinate system Σ. The outside magnetic field is in z-axis. The dielectric constant tensor of this plasma is given as

where

, ,

n is electron density and the frequency of incident wave. When the frequency is low, Raleigh criterion is valid, it is so approximately think that the magnetized cold plasma sphere locates in the electrostatic field^{ }[1,3]. The plasma has not electric charge in whole. According to, and considering that the differential of potential u is not relative to the differential order for x and y, the potential differential equation is obtained in the primary coordinate system as

Now, a scale coordinate system Σ’ is introduced as a new coordinate system. The coordinates of this system are indicated with x′, y′ and z′. The relation of coordinates between the two systems is written as

The differential equation of the potential in the scale coordinate system is derived by substituting the above expressions into Equation (2) and using the condition u = u′ [12,13] at any spatial point, Equation (2) may be expressed as

The condition u = u′ is understandable, for the potential is defined as the work done by the electric field to move a unit charge from one point to the reference point, namely W/q, so both the numerator and the denominator are scale invariants. Equation (3) shows that a magnetized cold plasma sphere in the primary coordinate system is transformed into an isotropic sphere in the scale coordinate system. This manipulation may greatly simplify the electromagnetic scattering problems.

The solution of Equation (3) can be obtained by using the method of separation of variables as follows:

Equation (4) is a general solution in the scale coordinate system. The parameters in the two coordinate systems are related, their relationships [12,13] are

In Equation (4), the term that may produce a finite potential in the sphere centre is considered. It is concluded that the expression of potential in the primary coordinate system can be obtained only by substituting the relations above into Equation (4). By utilizing the relation between D and E and the relation between the vectors in right angle system and spherical system, we may also obtain the expression of the dielectric constant tensor in spherical coordinate system as

where

We suppose that E_{0} is the magnitude of incident electric field, and that θ_{0} and φ_{0} are its directional parameters in the primary coordinate system. As shown in ^{ }[

When，the electric potential near to

Comparing the coefficients of the above expression with those in Equation (6) yields

where

Equations (4) and (6) are the electric potentials inside and outside the magnetized cold plasma sphere respectively. On the surface of the sphere, the electric potential inside the sphere is equal to that outside the sphere and the electric displacement D_{0} is continuous in the normal direction, namely

Inserting Equations (4)-(6) into the above conditions yields

There are three types of trigonometric functions in the above expressions, namely

Comparing their coefficients, we may obtain the following matrix equation

where

The solution of Equation (8) is easy derived as

Namely

We thus obtain the solution of electric potential inside and outside a magnetized cold plasma sphere as

From Equation (6) it follows that when θ_{0} = φ_{0} = 0, the incident electric field E_{0} is in the z-direction, B = D = 0，and A = –E_{0. }If we suppose that，now the problem of the electric field in a magnetized cold plasma medium has been changed into a question in the isotropic medium. Equations (10) and (11) are transformed, respectively, into the following expressions:

From [3,4,14], we obtain the solutions of an isotropic dielectric sphere in electric field E_{0}. These solutions may be given as

It can be seen that the results are consistent entirely with those in the literature. The correctness of the obtained results is therefore tested. Let and be the angles between E_{0} and the x-axis and between E_{0} and the y-axis, respectively, then it will be easily proved that. So the second term and the third term in Equation (10) are the potential produced by the polarizing electrical dipole moment respectively in x-direction and in y-direction. In the scattering of small particles, for example, Raleigh scattering, the electric field inside the target is of great importance. So we must discuss the distribution of the inside electric field in detail. The electric field is obtained by making a gradient from Equation (10) and utilizing the transformation between a vector respectively in the spherical coordinate system and the right angle coordinate system as follows:

Equation (12) demonstrates that the electric field inside a magnetized cold plasma sphere is a uniform field which is a complex function of the incident azimuth angle, outside magnetic field, the electric density and the frequency etc. This field makes an anglewith respect to the incident field E_{0}. This cosine function for this angle is easily derived by taking the scalar product of vectors as follows:

where

It is a function of the azimuth angle. The displacement D is easy obtained from Equations (1) and (12), it is not presented here in detail. Followings are partial numerical results:

According to the literature [^{17} m^{-3}. It is con-

cluded from ^{17} m^{-3} and outside magnetic fields B_{0} are respectively 0.004T and 0.005T. Another density of n = 8 × 10^{17} m^{-3} is used in the second dotted line. It obviously demonstrates that the outside magnetic field has not a great influence on the angle and however the electron density has a great effect on it. The angle is proportional to the frequency. It is well known that in isotropic medium, the angle is zero, so in the time varying electromagnetic field, the electric charges, negative and positive, in the cold plasma can not agreement with outside field as the frequency being augment which causes the angle’s accretion.

In this paper, the electric fields inside and outside a magnetized cold plasma sphere are investigated. We use the scale transformation theory of the electromagnetic field to reconstruct the Laplace equation and then obtain two analytical expressions of the electric potentials inside and outside the magnetized cold plasma sphere in detail. The obtained results are consistent with those in the literature when the dielectric constant tensor becomes that in an isotropic medium. The angle between the total fields inside and outside the magnetized cold plasma sphere is derived. The effects induced by the incident direction, outside magnetic field, frequency etc. on the direction and the magnitude of the inside electric field are simulated. Due to many particles such as sand-dust storm particulates, atomy particles and raindrops are generally anisotropic, so the results obtained can provide

a good theoretical foundation for studying the scattering features of small particles and magnetized cold plasma. How to use the scale transformation theory to study the electromagnetic fields inside and outside a magnetized cold plasma target irradiated by the time-varying electromagnetic wave is our next research subject.

This project was supported by the National Natural Science Foundation of China (Grant No. 60971079, 60801047), the Natural Science Foundation of Shaanxi Province (Grant No. 2009JM8020) and Natural Science Foundation of Shaanxi Educational Office (Grant No. 09JK800).