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Equipotential mapping

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It is not easy to measure or map the electric fields (E) in space. This experiment was done to provide a means of producing and mapping electric fields in uniform conductor, which can have the same form as an electric field in free space when the electrodes shapes and potential are the same.
  Experiment ED2; Equipotential Mapping.  Introduction; It is not easy to measure or map the electric fields (E) in space. This experiment was done to  provide a means of producing and mapping electric fields in uniform conductor, which can have the same form as an electric field in free space when the electrodes shapes and potential are the same. BRIEF THEORY.   We can base our experiment on our knowledge of Gauss’ law. According to Gauss law, the total flux of electric fields (E) out of any closed surface is e qual to 1/ε o times the total charge within the surface: in different form this may be written as; Div E =ρ/ε 0 ……………………………………..(1)   Where the charge distribution at any is point and is the permittivity of free space. Equation (1) applies to the medium in which the charges are fixed. (b). in free space, where there is no charge, equation 1 becomes, Div E =0……………………………(2)  (c). in a conductor in which the charges are free to move, we obtain the current density, J =σE……………………………….(3)   Where σ is the conductivity o f the conductor. Under steady conditions, we have Div j = 0 …………………………(4)  Combining equations (2), (3), and (4)., we get Div E =(1/σ)=div j= 0 …………………… (5)  (d). in any medium the feld is related to potential (v) by E = - grad V ……………………………(6)    By substituting (2), we get Div grad = ∆ 2   V = 0 …………………… (7)  Equation 7 is valid for both a conductor in which the steady currents are flowing in free space. We thus have the same differential equation to solve for the condition of steady current flowing  between the electrodes and the free space. The differential equation above (∆ 2 V = 0) is commonly known as Laplaces’ equat ion. It has one solution for each set of boundary conditions. Thus if we keep the boundary conditions the same, we may examine the EQUIPOTENTIAL surfaces between the electrodes and the free space by performing experiments on steady current flow between the same electrodes in a uniform conducting medium, e.g., water. In the following experiment only two dimensional cases were considered . The conducting medium was water with uniform depth in a shallow tank. The electrodes are perpendicular to the water surface. The equipotential were considered as equipotential lines on the water surface. The use of steady potentials and fields with water (or any other electrolyte) would be in the long run give trouble with the polarization and chemical decomposition at the electrodes. To overcome this, we approximate to the steady states by using low alternating voltages. Even at the frequency of 2 KHz the inhomogeneity caused by charges moving together and separately in the water is negligible and still the steady state condition is effectively satisfied. In this experiment, the potentials are mapped directly by balancing them against potentials  provided by a Rayleigh potentiometer, using earphones as a null detector. The principle is similar to that of a Wheatstone bridge. THE EXPERIMENT SET-UP DIAGRAM.       R1 and R2  are two decade resistance boxes making the Rayleigh potentiometer, such that R1 + R2= 1000Ω  always.      The probe is a bare wire, or some of pointer, that is moved through water between the electrodes to determine the position at which the ear phones detect minimum signal.    If we regard the first and the second electrode to as being V 0  volts potential (the output of signal generator). Then the potential at the probe is V = R  2   V 0 / (R1+R2)   EXPERIMENT   PROCEDURE ; 1.   The tank was dried and then the tank was placed onto a sheet of graph paper which was to provide a co- ordinate system. 2.   Small amount of water was put to level the bottom of the tank with wedges, then filled it uniformly to a level of about 1 cm. 3.   The two long straight electrodes were arranged parallel to one another in the middle of the tank, their distance between them being about two thirds of their lengths.  4.   The circuit was connected as shown in the circuit diagram. The probe would be moved along a line such that the sound in the earphones was always a minimum. This line is therefore the equipotential. 5.   A table of values of V,R  1 , and R  2  used was made. The value of the voltage used gave the  best signal. 6.   The first pair of values: R1 and R2 were set and the end of the probe dipped in the water. It was moved until minimum signal sound was heard in the headphones. The corresponding point was marked to scale on the map. Other points were located in the same way, their numbers sufficient to draw contour for the particular value of V .  7.   Other values of R1 and R2 were taken in turn and a complete set of contours or equipotential lines drawn.  8.   Similarly one electrode was replaced with a pin electrode, dipped into the water and the equipotential contour mapped for this arrangement.  9.   The process was repeated using a circular strip electrode.   After the experiments with the given sets of the contours corresponding to the electrode configurations, the field lines for each configuration was drawn from the background knowledge of relation between electric field and potential. The lines were made continuous from one electrode to another while fulfilling the necessary conditions at the electrodes. The condition; The field lines should be perpendicular to the electrodes (the equipotential lines).  THE DATA COLLECTED WAS AS FOLLOWED; (a)   Using straight electrodes; R  1  R  2  V 0  CO-ORDINATES 500 500 5 (1,8) (1,5) (1,3) 600 400 5 (-2,8) (-2,5) (-2,3) 700 300 5 (-4,8) (-4,5) (-4,3) (b)   Using circular electrodes;
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