The electrostatic field is determined by the charge distribution. Given the charge distribution, medium distribution and boundary conditions, the electric field distribution can be obtained according to Maxwell's agenda group and boundary conditions. But in most cases, the analytical solution is obtained, so it is necessary to find the electric field distribution by numerical solution or experiment.

Experimental purpose

1. Learn to describe and study the distribution of electrostatic field through simulation.

2. Master the conditions and methods of simulation application.

3. Deepen the understanding of basic concepts such as electric field strength and potential.

Laboratory instrument

Conductive liquid electric field plotter, coaxial electrode, parallel plate electrode, white paper (self-provided)

Experimental principle

It is very difficult to measure the electrostatic field directly, because the instrument (or its probe) will change the measured electric field to some extent. If it is measured by an electrostatic instrument, it is not possible because there is no current flowing in the field. Therefore, the constant current field is used to simulate the electrostatic field in the experiment. That is, the corresponding electrostatic field distribution is mapped by mapping the distribution of the constant current field at the point.

The requirement of simulation method is to simulate a field (called simulation field) so that its distribution is completely consistent with that of electrostatic field. When the probe is used to detect the bending potential distribution, the electric field distribution will not be distorted, so the electrostatic field can be measured indirectly.

One of the methods of measuring electrostatic field by simulation method is to replace electrostatic field with current field. According to the electromagnetic theory, the current field of steady current in electrolyte (or water liquid) is similar to the electrostatic field in dielectric (or vacuum). In the passive region of the current field, the physical laws followed by the current density vector and the electric field intensity vector of the electrostatic field have the same mathematical form, so the two fields are similar. Under similar field source distribution and similar boundary conditions, their solutions have the same mathematical model. If two electrodes connected to a power supply are placed in a poor conductor such as a dilute solution (or aqueous solution), a current field will be generated in the solution. There are many equipotential points in the current field, which are measured and described as equipotential surfaces. These surfaces are also equipotential surfaces in electrostatic field. Usually, the distribution of electric field is in three-dimensional space, but when conducting simulation experiments in water and liquid, the measured electric field is in a horizontal plane. In this way, the equipotential surface becomes an equipotential line, and the power line can be drawn according to the orthogonal relationship between the power line and the equipotential line. The tangent direction of each point on these power lines is the direction of electric field intensity at that point. This can use equipotential lines and power lines to visually represent the distribution of electrostatic field.

When detecting the equipotential point in the current, the distribution of the current line will not be affected, and the measuring branch cannot take out the current from the current field. Therefore, it is necessary to use high internal resistance voltage to eliminate this influence. When the electrode is connected with AC voltage, the instantaneous value of the generated AC electric field changes with time, but the effective value of AC voltage is equivalent to that of DC voltage (see Appendix), so the equipotential line for measuring the effective value in AC electric field with AC voltmeter is exactly the same as that for measuring the same value in DC electric field.

The application condition of simulation method is that the basic law of "simulation field" or the mathematical agenda it meets is exactly the same as that of the simulated field. This simulation is mathematical simulation. Constant current field and electrostatic field satisfy similar partial differential equations, as long as the shapes and sizes of charged bodies (i.e. electrodes), their relative positions and boundary conditions are the same. Then the distribution of these two fields is the same.

According to the corresponding relationship between electrostatic field and constant current field, electrostatic field can be simulated by the following constant current field: two long straight coaxial cylindrical conductors, with inner cylinder radius a and outer cylinder radius b, filled with dielectric with uniform capacitance ratio, and the inner and outer cylinders keep potential difference v0 = VA-VB. As long as the distribution of simulated constant current field is measured, the distribution of simulated electrostatic field can be obtained.

Electrostatic fields of different shapes can be simulated without electrodes of different shapes, such as parallel plate electrodes, and electrostatic fields in parallel plate capacitors can be simulated.

Figure a figure b

As shown in Figure A, it is a coaxial cylindrical electrode, with inner electrode radius a, outer electrode radius b, inner electrode potential Va and outer electrode potential VB = 0, and the potential on the distance r axis between the two electrodes is:

,

According to Gauss theorem, the electric field on a cylindrical surface with radius r is:

Where λ is the electric quantity per unit length of cylindrical surface and ε is the dielectric constant between two poles, which can be obtained by two formulas.

When r = b, substituted into the above formula are:

This formula is the potential distribution formula in the electrostatic field between coaxial cylindrical electrodes.

If a uniform bad conductor is filled between coaxial cylindrical electrodes, a stable current field will be formed between the electrodes. Similarly, the potential distribution formula in the steady current field can also be deduced as follows.

Comparing these two formulas, it is not difficult to see that they both satisfy the Laplace equation of Gauss theorem and their potential distributions are the same. The stable current field will not be distorted by the introduction of the probe, so the stable current field with the same electrode size and the same boundary conditions can be used to simulate the electrostatic field for detection, thus indirectly depicting the distribution of the electrostatic field.

Experimental contents and steps

1. Connect the lines according to the circuit diagram (Figure B shows the coaxial cylindrical electrode).

2. Level the bottom of the sink frame with a level. Inject a certain amount of water into the water tank, paste white paper on the frame of the water tank, and record the mapping points; Turn on the power supply, adjust the voltage to 10V, read the value when the digital voltmeter is set to "output", and put the probe outside the water tank.

3. The probe is in close contact with the internal electrode, and the voltage is displayed as 10V, which is read when the digital voltmeter is set to "detect". If the voltage is 0V, change the output polarity of the power supply voltage.

4. Let the probe move slowly between the poles, and measure equipotential lines with voltages of 7.0V, 5.0V, 3.0V and 1.0V in turn, with 8 measuring points for each equipotential line.

5. Take three recording points and one recording point along the inside and outside of the external electrode with a probe to determine the electrode center and the thickness of the external electrode; Record the diameter of the inner electrode and the inner diameter of the outer electrode.

6. Repeat the above two steps (2 and 3) with parallel plate electrode instead of coaxial cylindrical electrode, and record eight measuring points (evenly distributed) along three equipotential lines of 7.5V, 5.0V and 2.5V respectively, so that the measuring points can determine the electrode position.

7. Mark the measuring point and equipotential line value of each equipotential line with different symbols on the parallel plate electrode measuring paper, draw the electrode, and draw the experimental equipotential line and electric field line.

8. On the coaxial cylindrical electrode recording paper, determine the center of the circle by geometric method, draw the inner and outer electrodes, mark the measuring point and equipotential line value of each equipotential line with different symbols, and draw the theoretical equipotential line (calculated according to the formula) and electric field line.

9. Measure the distance from each measuring point of equipotential line on coaxial cylindrical electrode recording paper to the center of the circle, and calculate the average value. Draw the theoretical curve of VR/VA ~ LNR on semi-logarithmic coordinate paper, mark the corresponding experimental measurement points, and draw the experimental curve.

Key points of experimental teaching guidance

1. The simulated field not only satisfies the mathematical equation and boundary conditions similar to those of the fringe field, but also requires that the bottom of the water tank must be horizontal, the conductivity of the solution is much smaller than that of the electrode and uniform everywhere, and the power supply must be alternating current with a certain frequency to prevent dielectric polarization.

2. The water in the water tank should not overflow the upper surface of the electrode.

3. The wires must be firmly connected to avoid the output voltage failing to meet the requirements due to contact resistance.

4. The electric field line should start from the outer surface of the high potential electrode and end at the inner surface of the low potential electrode, and be perpendicular to the equipotential line everywhere. The density of electric field lines reflects the strength of electric field.

5. When the upper recording paper is spotted, don't use too much force, just press it gently to avoid moving the electrode and bringing errors.

6. When doing experiments, determine the center of the circle; Determine that position of the electrode; In addition, the edge effect of the area extending outward between two plates should also be described.

7. When drawing, you should not only draw equipotential lines, but also draw electric field lines (starting with positive charges and ending with negative charges).

a

b

Electric field distribution of coaxial cylindrical electrode

Electric field distribution of parallel plate electrode

10.0v

7.5V

5.0V

2.5V

0.0V

8. When drawing on semi-logarithmic coordinate paper, theoretical straight line and experimental straight line should be made at the same time.

Vr/va ~ lnr curve

Virtual reality/virtual reality

1.0

0.8

0.6

0.4

0.2

0.3 0.4 0.6 0.8 1.2 3 4 5 6 7 89 cm

This graph is drawn with semi-logarithmic graph paper, which is evenly distributed in the longitudinal direction and logarithmically distributed in the transverse direction. The theoretical curves in the figure are (ln a, 1) and (ln b, 0), and the four points on the line are measured points.

9. The electric field formed when DC voltage is applied between electrodes is time-invariant and stable. If an alternating voltage is applied between the electrodes, the electric field will change with time and the scene will be stable. However, considering that different DC voltages are applied between the straight electrodes, the geometry and position of equipotential lines or surfaces with a certain relative potential value in the electric field remain unchanged. When AC voltage is switched on, it can be regarded as block speed conversion of DC voltage (including polarity) with different values on the electrode. Therefore, the starting point and position of the relative potential line measured in alternating current field and the equivalent relative potential equipotential line measured in direct current field are exactly the same. However, it must be pointed out that the exact same here is conditional. The frequency of alternating current we choose should not be high, otherwise the point potential between electrodes will not increase or decrease at present. We choose 50HZ alternating current, and the influence of electrodes and bad conditions on each other's capacitance can be ignored. Fully qualified.

Ask questions immediately in the experiment.

1. Question: What method is used to draw the electrostatic field in this experiment? Why use this method?

A: The electrostatic field is drawn by simulation. Because the distribution of electrostatic field is directly measured, it is necessary to measure every point in the space point by point with a probe. When the probe is put into an electrostatic field, the induced charge will be generated on the probe due to electrostatic induction, which will change the charge distribution of the original electrode and cause the distortion of the original electric field. Obviously, direct measurement is not feasible, so simulation method is used for surveying and mapping.

2. Q: What are these two simulation methods and what are their application conditions? What kind of simulation is used in this experiment?

A: Simulation methods are divided into physical simulation and mathematical simulation. The application conditions of physical simulation are physical similarity and geometric similarity, that is, both the model and the prototype obey the same physical laws; The geometric size of the model is enlarged or reduced in proportion to the geometric size of the prototype. The application condition of mathematical simulation is that the model and prototype can be completely different in physical essence, but they all follow the same mathematical law, that is, they satisfy similar mathematical equations, and the shapes and sizes, relative positions and boundary conditions of charged bodies (that is, electrodes) are the same. Using mathematical simulation method, the electrostatic field is simulated with constant current field.

3. Question: Why is the electric field distribution in a bad conductor the same as that in a vacuum?

Answer: Because when there is no current flowing in a bad conductor, the positive and negative charges in any macro-volume element are equal, and there is no net charge, so it is electrically neutral. When DC current flows, the charge flowing out of the volume element is replaced by the same sign charge flowing in per unit time, and the number of positive and negative charges in the volume element is still equal, so the whole volume is electrically neutral. In other words, the electrostatic field in vacuum is generated by the charge on the electrode, and the electric field is also generated by the charge on the electrode in the bad conductor through which the constant current flows. The difference is that the charge on the electrode in the electrostatic field is static, while the charge on the electrode in the constant current field is lost at any time and replenished by the power supply, and the amount of charge remains unchanged in the dynamic equilibrium state. Therefore, the distribution of electric field is the same in both cases.

4. Question: What are the experimental conditions for simulating electrostatic field with constant current field?

Answer: The experimental conditions first require that the conductivity of the bad conductor be kept constant in the area between the two electrodes and its thickness be kept constant; Secondly, there is basically no current flowing through the instrument that measures the potential. Essentially, it is to ensure that the potential distribution of the constant current field in the area and boundary between electrodes will not change due to the measurement operation.

5. Question: How to make measuring points in the experiment?

Answer: (1) The coaxial cylindrical electrode is used as the measuring point along the radial direction.

Firstly, measuring points are made along two radii perpendicular to the experimenter, and four measuring points are made in turn from high potential to low potential (from central electrode to external electrode) along the radius; Make four measuring points along another radius line in the opposite direction; Secondly, make measuring points along two radii in the horizontal direction, and finally make four radius lines inclined left and right, whose layout is like a "meter". The advantage of this is that the measuring points will not be missed, and the measuring points on the same equipotential line can also be evenly distributed. After all the measuring points on the equipotential line are completed, three measuring points should be made along the outer edge of the external electrode to determine the center of the electrode.

⑵ Use the parallel plate electrode as the measuring point along the equipotential line, and make the equipotential line from high potential to low potential in turn.

When making measuring points along the equipotential line, it is not limited to the area between the two ends of the electrode, but must extend outward. Because the length of the parallel plate electrode in the experiment is limited, the electric field at both ends of the electrode has edge effect, so four measuring points are made in the middle along the equipotential line, then one measuring point is made at both ends of the electrode, and another measuring point is made at both ends about 1 cm. Finally, a measuring point is made at each corner of the two electrode plates to determine the position of the electrodes.

6. Question: How to draw the electric field distribution map?

Answer: (1) Coaxial cylindrical electrode:

Firstly, according to the three measuring points of the outer edge of the external electrode, the center of the circle is determined by geometric drawing method, and a complete coaxial cylindrical electrode is drawn from the measured radii A and B. The theoretical radius of each equipotential line is calculated by formula, and the equipotential line is drawn according to the theoretical radius. Mark eight measuring points on the same equipotential line with the same symbol, and indicate the size of the equipotential line. Then, according to the characteristics that the electric field line and the equipotential line are orthogonal everywhere, draw the electric field line and mark its direction (from the formula E=-dU/dr, the direction of the electric field is from high potential to low potential).

Because in the state of electrostatic balance, the electric field inside the conductor is zero, and the charge is distributed on the surface of the conductor. Therefore, the electric field lines start from the outer surface of the center electrode and end at the inner surface of the outer electrode.

Finally, write the name of the picture (electric field distribution of coaxial cylindrical electrode).

(2) Parallel plate electrode

According to the measuring points that determine the electrode position, draw two electrode plates, respectively connect eight measuring points with the same potential with smooth curves, draw equipotential lines, mark the measuring points with the same symbols, and indicate the values of equipotential lines. Draw electric field lines and their directions like coaxial electrodes. When drawing electric field lines, special attention should be paid to the edge effect of the electric field near the two ends of the plate. Write the picture name (electric field distribution of parallel plate electrodes).

7. Q: Why do you want to draw a cylindrical electrode with a logarithmic drawing paper? How to draw drawing paper with simple logarithm?

Answer: According to the formula of the inter-electrode potential, it is only a function of the coordinate R, so the logarithmic graph paper can express a linear relationship with R, which is very convenient to draw.

Logarithmic graphic paper (also known as semi-logarithmic graphic paper) takes the logarithm of an axis when making it. From the coordinate paper, the scale values are evenly distributed on one axis, and the logarithm is distributed on the other axis. When drawing "~ theoretical curve" diagram of coaxial cylindrical electrode, draw a theoretical straight line with evenly distributed scale as vertical axis, logarithmic distribution scale as horizontal axis, and point (lna, 1) and point (lnb, 0) as endpoints. Then measure the actual radius of eight measuring points on each equipotential line and record it in the data table, calculate the actual average radius value of each equipotential line, and trace the average point on the coordinate paper to see if it falls on the theoretical straight line.

8. Question: Can you simulate the electric field of a parallel axis or a parallel long straight cylinder with the same charge amount and different signs? Why?

Answer: Yes. According to the electromagnetic theory, the current field of steady current in electrolyte (or water liquid) is similar to the electrostatic field in dielectric (or vacuum). In the passive region of the current field, the physical laws followed by the current density vector and the electric field intensity vector of the electrostatic field have the same mathematical form, so the two fields are similar. Under similar field source distribution and similar boundary conditions, their solutions have the same mathematical model. If two electrodes connected to a power supply are placed in a poor conductor such as a dilute solution (or aqueous solution), a current field will be generated in the solution. There are many equipotential points in the current field, which are measured and described as equipotential surfaces. These surfaces are also equipotential surfaces in electrostatic field. Usually, the distribution of electric field is in three-dimensional space, but when conducting simulation experiments in water and liquid, the measured electric field is in a horizontal plane. In this way, the equipotential surface becomes an equipotential line, and the power line can be drawn according to the orthogonal relationship between the power line and the equipotential line. The tangent direction of each point on these power lines is the direction of electric field intensity at that point. This can use equipotential lines and power lines to visually represent the distribution of electrostatic field.

9. Question: Please give the main steps to simulate the electric field between a flat plate and a long straight charged cylinder on its vertical plane?

Answer: (1). Connect the lines according to the circuit diagram.

(2) Use a level to level the base of the water tank rack. Inject a certain amount of water into the water tank, paste white paper on the frame of the water tank, and record the mapping points; Turn on the power supply, adjust the voltage to 10V, read the value when the digital voltmeter is set to "output", and put the probe outside the water tank.

(3) The probe is in close contact with the internal electrode, and the voltage is displayed as 10V, which is read when the digital voltmeter is set to "detect". If the voltage is 0V, change the output polarity of the power supply voltage.

(4) Let the probe move slowly between the two poles, and measure equipotential lines with voltages of 7.5V, 5.0V and 2.5V in turn, with 8 measuring points for each equipotential line.

(5) Take three recording points along the outer side of the charged cylindrical electrode with a probe and four recording points along the outer side of the charged flat electrode with a probe to determine the electrode center and electrode thickness.

(6) Mark measuring points and equipotential lines with different symbols on the measuring paper, draw electrodes, and draw experimental equipotential lines and electric field lines.