3.8.3 Experimental principles
The electrostatic field is induced by the distribution of the static charges.In principle,electric field strength E(a vector quantity)and electric potential U(a scalar quantity)can be employed to describe the electrostatic field.The quantity of the vector E is determined by the gradient of the potential,with the vector equation E=-∇U.To visualize the magnitude and direction of an electrostatic field,the concept of electric field lines is introduced.Here,the electrical field line is defined as the path of a free charge that moves from one position to another in the electrostatic field.However,it is difficult to accurately measure the electrostatic field based on the following two reasons:
①It is impossible to measure the electrostatic field by simple magnetoelectric instrument because there is no current in the electrostatic field.
②The testing probe would perturb the original electrostatic field once it is placed in the electrostatic field.Within this context,the“imitative method”has been utilized to map the electric field in this experiment.
In this experiment,the steady current field is employed to simulate the electrostatic field since the two fields follow the same physical laws in the mathematical form under certain conditions.According to the electromagnetic field theory,the spatial distribution of the steady current field is the same as that of its corresponding electrostatic field.In other words,an equivalence between the electrostatic field and steady current field could be obtained under the condition of the same electrode shape,the electric potential at the electrode and uniform space medium,that is,Usteady current field=Uelectrostatic field or Esteady current field=Eelectrostatic field.
Figure 3.8.2 The electrosatic field of a coaxial cylindrical cable
Now,the electrostatic field of a coaxial cylindrical cable and its corresponding steady current field have been used to discuss the equivalence mentioned above in detail.As shown in Figure 3.8.2,the cylindrical conductor A and the cylindrical shell conductor B are placed in the same axis with equivalent charges but the opposite sign,respectively.In addition,there is a vacuum between A and B.The radius of A and B are R1 and R2,respectively.Set the potential of the outer cylinder as zero and that of the inner cylinder as U0.According to the Gauss theorem,the electric field lines radiate from A to B,and the equipotential surface is a group of coaxial cylindrical surfaces.Therefore,we only need to focus on the electric field distribution on the cross section P of any vertical axis.As shown in Figure 3.8.2,the electric field strength at any point with the radius r is:
Where λ is the linear density of charge at A(or B).The potential at the radius r is:
In Equation,if r=R2,then UR2=0.Therefore,this Equation can be obtained.
By substituting(3.8.2)formula,we can get the potential at the radius r,which can be expressed as:
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The field strength at the radius r can be expressed as:
If the space between A and B is filled with a poor conductor(its resistivity is ρ),and A and B are respectively connected with the positive and negative electrodes of the power,as shown in Figure 3.8.3(a).At that time,a radial current is formed between electrodes A and B,and a steady current field is established.In the same way,we can use the coaxial cylinder slice with the thickness of δ to conduct researches.If the radial resistance of the cylindrical piece from radius r to r+dr is,then the total resistance of the conductor from radius R1 to R2 is:
The conductor resistance between radius r and R2 is:
The radial current is:
The potential at the radius r is:
It can be seen that Equation(3.8.9)and Equation(3.8.4)have the same form,indicating that the potential distribution of steady current field and electrostatic field is the same.The analysis above shows that the equivalence is achieved between the electrostatic field and the steady current field.Therefore,in order to depict the distribution of the electrostatic field,we only need to map the electric field distribution of the corresponding steady current field.
Actually,the electric field around the electrode is spatially distributed,and the equipotential surface is a group of disjoint surfaces.For the sake of simplicity,this experiment only focuses on the electric field distribution on the cross section,as shown in Figure 3.8.3(b).
Figure 3.8.3 Simulation diagram of electric field distribution between parallel transmission lines
