Thus the charges follow the field lines and the current density is proportional to intensity of the field.

Transcription

1 FYSA 0/ K1 EQUIPOTENTIAL SURFACES In this laboratory work we examine the potential distribution created by the electric field of various conductors having different potentials. The task is to find equipotential surfaces (D equipotential lines) and draw a picture of the conductor system and equipotential lines. With the help of the picture we can do conclusions about the electric field lines and intensity of the field. Equipotential surfaces are calculated also numerically. 1 BASES o through chapter 3 Electrostatic field calculations in rant & Phillips, nd edition. Numerical solutions are handled in section 3.5. Do the home work before coming to the shift. 1.1 The electrolyte basin When two perfectly conductive electrodes are placed in medium which is infinite wide and which have finite conductivity properties, the potential distribution is similar to the case where electrodes are forming capacitor and the medium would be insulator. This follows because boundary conditions are equal (surfaces of the electrodes are always equipotential surfaces). For example two conductors, charges q and q, (the voltage between V=q/C) in vacuum induce the electric field which is the same as if the medium would be the conductive electrolyte and the voltage between conductors is held constant (V) by the external voltage source. In the latter case the current flows in the direction of the electric field j σe. (1) Thus the charges follow the field lines and the current density is proportional to intensity of the field. Based on this we can measure the potential distribution of the electrostatic field from corresponding electric flux field. If the symmetry of the system allows, that we can form the conception of the field already from D distribution of the potential, we can do measurements in the basin filled with the electrolyte. Surfaces of the electrodes are along z-axis. The finite size of the

2 basin produces some distortions to the distribution of the potential. If the basin is insulating material the equipotential lines try to turn against the edges. 1. The relaxation method In the region where there are no charges Laplace s equation holds 0. () In the system consisting of the electrodes that have different potentials and lie in uncharged medium, the potential can be solved by solving the Laplace s equation. The boundary conditions are that the potential on the surface of the electrodes is constant. Usually this is difficult to solve analytically. Numerical solution is quite easy to find at least by using the computer. Let s look at the D case. We can assume that the potential doesn t depend on z, so the Laplace s equation gets the form 0 x y. (3) We can consider a region in x-y plane, bounded on two equipotentials at 0 and 10 volts (Fig. 1). The field lines at the edge are parallel to edge. The task is to determine the potential elsewhere. The solution is trivial because the potential changes linearly from 0 V to 10 V but we solve it numerically. Let s study the potential around point.

3 Figure 1. Equally spaced points used in finding an approximate solution to Laplace s equation. Distance of the points is h. If the potential and its derivates are known at the potential at the neighbouring points F, H, C and K can be calculated by the Taylor series. F x h x h 1 H x h x h 1 C y h y h 1 K y h y h 1 If the grating is dense enough (h is small), the higher order terms can be ignored. Adding the equations above and using Laplace s equation we find

4 1 C F H K. (4) 4 Eq. 4 isn t exact, because we have ignored third and higher order terms. By decreasing the distance between points the fault can be diminished. We know the exact values of the potentials at the edge points so we can use those in numerical calculations. This way we manage to avoid calculating the potential of the edge points. If we wanted to calculate the potential of the edge points we need to know the potentials outside the region. At first assume that the potential is zero at points A-P. Let s start using eq. (4) at point A 1 A 0 E B For the same way 1 B 4 A F C. By using f A = 0.5 we get f B = 1.15 and so on. All points go through and then start over. The whole operation is carried out over and over again until eq. 4 is satisfied at every point on the grid. In this example, the method converges to within 1% of the exact solution after 18 iterations. This method can also be used in 3D. In all cases the Taylor expansion leads to approximate relations between the potentials at neighbouring points. The relaxation method can be excited by so called over relaxation. ( n) (n 1) r (calc.) (n 1), where the f(calc.) comes from eq. 4 and r is the relaxation factor (1 r ). When r = 1 the relaxation is normal. Typical start-up value for r is but when relaxation proceeds it s conventional to decrease the value of r to 1. Too big relaxation factor leads to oscillation of relaxation and instability. The other way to excite relaxation is to calculate the potential only at the every second points. Omitted points are taken into account later. More spacing than one can be

5 done but then there is a risk that whole electrode is uncounted, if the electrode is consisted of only few grid points. The potential is calculated above by using the latest value for the neighbour points. The simplest way would be to use so called Jacobi method where to the potentials in the iteration round n is calculated from values from round n-1. However, this is quite slow method. So called auss- Seidel method is much faster. You can read more about relaxation from Numerical Methods For Engineering Application by Joel. H. Ferziger. Home work: For understanding the relaxation method, calculate few iteration rounds for 1 dimensional system. The points at the edge are in fixed potential V and V. You can use for example 5 grid points. Compare the values you get when using a.) values from the latest round n-1 b.) the latest value ( round n or round n-1 depending order of calculation) How does this affect the propagation velocity of relaxation?

6 THE EQUIPMENT The electrolyte basin consists of the transparent, flat-bottom acrylic box filled with tap water. Water is the electrolyte. Do the measurements in the middle of the box so that the distortion from the edges is as little as possible. The system is shown in figure. Figure. The measurement system The measurement is basically the bridge test where the sliding control of the potentiometer is set to the wanted potential. The equipotential lines are searched by the probe. Move the probe across the basin keeping it perpendicular to the bottom. When the probe is in the same potential as the potentiometer (the equipotential line) there goes no current through the galvanometer.

7 3 MEASUREMENTS Couple the potential difference (about 5 volts) between the electrodes. Find the equipotential lines that are 0.1 * V apart from each other (V = the potential difference). Start from 0.1 V. Notice, that the absolute value of the voltage isn t important because of the bridge measurement. Find also the potential of the separate conductor if it exists. Moving the probe along the equipotential line you can look for the x-y-coordinates of the probe and then draw the figure of the lines (Fig. 3). Bring your own graph paper with you! Figure 3. An example of the equipotential surfaces.

8 1) Plot the electrode system and corresponding equipotential lines as follows: a) The conducting ring in homogeneous electric field: At first notice that the field is homogeneous when the conducting ring isn t in the basis. What is the potential inside the ring? b) The dipoles c) The field between electrode and the tip

9 ) Measure the potential between two cylindrical conducting bodies as a function of radius r (distance from midpoint). Find r for 10 different voltage values and make a plot (ln r, V) (ln is the natural logarithm). Check are the results in accordance with the equation r 1 E A and U(r) Ed r Aln(r) B, r r 1 where A and B are constants.

10 3) Use the Equipotential-software to calculate the distributions of potentials in sections 1b) and 1c). If you have no time for this during the shift you can do this part some other time later and return the papers to the student laboratory once finished. 4 THE EQUIPOTENTIAL SOFTWARE The Equipotential software (short-cut tasapot-rd) is installed in computers of ADP-class and also in student laboratory s computers. The software resolves the Laplace s equation in region of 40 cm * 8 cm. The edges of this area are assumed to be insulating material. The grid points are at intervals of 0.5 centimetres. You can place at most 100 rectangular and at most 100 circular electrodes. You have to specify the potential to every electrode. You can proceed ahead step by step. In every step you can give the number of the iterations, the relaxation factor and the number of the spacing. You can also go through the points in two directions (from left bottom corner to right top corner and vice versa). In that case the number of iterations is doubled. After every step the programme reports the residual and the biggest variation in the potential between two last loops. The residual ( n) ( n1) i i i. ( n1) i i If you want to continue, the programme asks again the number of iteration, the relaxation factor and the number of the spacing. User can change the calculation methods during the calculation. There aren t specific manual for the programme. It s interactive and asks all the information needed. Unfortunately all questions are in Finnish. Ask from the assistant if you have any problems! When you open Tasapot-rd it first asks you to enter the user name. You can put for example your initial letters as user name. Then you have to put password. After that opens a window

11 where are told facts about the relaxation factor. Then appears a window where are told about the spacing and the electrodes. Now you can start calculations. There seven (7) options 0 Lopeta (Exit) 1 Määrittele elektrodit (Define the electrodes) Laske potentiaali (Calculate the potential) 3 Näytä potentiaalipinta (Show the potential surface) 4 Näytä tasa-arvokäyrät (Show the equipotential lines) 5 Tulostuksen esikatselu (Preview) 6 Tulosta kuvaajat (Print) Valintasi? > (Your choice?) > At first you should define the electrodes so you choose 1. Now the programme asks: 1.) Lopetetaanko, määritelläänkö ilmaa vai elektrodit ( l / i / E )? (Exit, define air or the electrodes? (l / a / E )) If you want to exit press L. If you want to define the electrodes press E. After that, you have to choose do you want to define rectangular or circular electrodes..) Elektrodin/ilman tyyppi Ympyrä/Suorakaide (Y/s)? (Type of an electrode/air Circular/rectangular (c/r)?) Select circular (press Y) or rectangular (press S). If you select Y: 3.a) Anna ympyrän keskipiste ja säde X0, Y0, R? (ive the midpoint of the circle and its radius X0, y0, R?) Lengths are in centimeters. If you select S: 3.b) Anna suorakaiteen reunat : X(ala), X(ylä),Y(ala),Y(ylä) (ive the edges for electrode: X(min), X(max), Y(min), Y(max)) Lengths are in centimetres. 4.) Anna elektrodin potentiaali? (ive the potential of the electrode?) 1.) Now you have defined one electrode but you have to define also another. So you can repeat steps 1.) - 4.) You have to change the lengths and potential. After you have defined two electrodes you

12 can select L, so you end defining the electrodes. Now is time for calculate the potential. So select Laske potentiaali. 5.) Lasketaanko iteraatio edestakaisin (k/e)? (Calculate the iteration back and forth (y/n)?) Press either k or e. 6.) Käytetäänkö vanhoja vai uusia tietoja (v/u)? (Using old or new information (o/n)? Select either v or u. 7.) Iteraatioiden lukumäärä? (Number of iterations?) ive some number for example between ) Relaksaatiotekijä, 1 r<? (Relaxation factor?) Select r=1 for all calculations. 9.) Harvennusten lukumäärä (0, 1, tai 3)? (Number of the spacing (0, 1,, or 3)?) Select 0. Now the programme has calculated the potential. You can either finish or continue. 10.) Jatketaanko laskemista (k/e)? (Continue (y/n)?) Select k or e. If you select k (yes) you repeat steps 5.) -10.). If you select e (no) the calculation ends and you can look the potential surface by selecting 3 and look the equipotential lines by selecting 4. In both cases you can close the graph window and return to control menu by clicking the right mouse button on the graph window. When you want to print those figures ask help from the assistant. Use the preview possibility as much as possible. You can also construct a hollow electrode by defining air. For example a ring which outer radius is 10.5 cm and inner radius 10.0 cm. At first you define a circle electrode with radius 10.5 cm and after that you define circle shaped air with radius 10.0 cm and the midpoint equal to the electrode. You have to define the electrode first!