Assume a magnetized cold plasma sphere to have radius R₀ 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 E₀ is the magnitude of incident electric field, and that θ₀ and φ₀ are its directional parameters in the primary coordinate system. As shown in Figure 1, since the outside electric potential is not symmetrical, we can write the potential as [2]

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 D₀ 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 θ₀ = φ₀ = 0, the incident electric field E₀ 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 E₀. 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 E₀ and the x-axis and between E₀ 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 E₀. 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 [15], the parameters used in simulations are f = 3 GHz，n = 4.5 × 10¹⁷ m^-3. It is con-

cluded from Figure 2 that the inner field will decrease when the outside magnetic field increase. The angle between the outside electric field and the outside magnetic field has a great effect on the field inside the plasma and further simulations show that angle has not affect on the inner field. This is because that the anisotropic property in plasma is caused by the outside magnetized field, the components of outside electric field are relative to the angle and angle has a good symmetry to outside magnetic field, so the inner field does not varying with angle. In Figure 3, the azimuth angles are. It is demonstrated that the inner electric field decrease as the electron density increasing. This is caused by the reason that when the electron density increases, the electrical conductivity is also strong, so the shield of electric field is enhanced. It is also can be seen from Figure 4 that the inner field is proportional to the operating frequency. This is due to the reason that the variety of electric charges inside plasma not agreement with that of outside electric field and the shielding effect is thus debased. Figure 5 had conducted the change of angle versus the frequency. The solid line and dotted line are nearly superposed in which the density n = 4.5 × 10¹⁷ m^-3 and outside magnetic fields B₀ are respectively 0.004T and 0.005T. Another density of n = 8 × 10¹⁷ 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.