Experimental Mapping of Electric Fields: Theoretical Foundation and Data Analysis

The goal of this laboratory experiment is to develop a method for producing and mapping electric fields in a uniform conductor, simulating conditions similar to those in free space. This experiment is based on Gauss' law, which relates the electric field to the charge distribution within a closed surface. The theoretical foundation involves equations (1) to (7), incorporating concepts like current density, conductivity, potential, and Laplace's equation.

Experimental Setup:

1. Materials:
   • Shallow tank filled with water (conductive medium)
   • Electrodes perpendicular to the water surface
   • Rayleigh potentiometer
   • Earphones as a null detector
   • Low-frequency alternating voltage source (approx.
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     2 KHz)

2. Procedure: a. Submerge electrodes in the water, ensuring uniform depth. b. Apply low alternating voltages to mimic steady states, minimizing polarization and chemical decomposition. c. Utilize the Rayleigh potentiometer and earphones to balance potentials and detect null points. d. Record data for various voltage configurations between electrodes.

Theoretical Background:

1. Gauss' Law: Div E=ε0​ρ​
2. Conductor Conditions: a. In free space: Div E=0 b. In a conductor with free charges: J=σE and Div J=0 c.
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   Combining equations: Div E=σ1​=Div J=0

3. Relation to Potential: E=−∇V
4. Laplace's Equation: ∇2V=0

1. Record potential values for different electrode configurations.
2. Map equipotential lines on the water surface.
3. Note the null points detected by the earphones, indicating balanced potentials.

Discussion:

1. Validity of Steady State:
   • Justification for using low alternating voltages.
   • Consideration of frequency (2 KHz) and its impact on inhomogeneity.
2. Comparison with Theoretical Models:
   • Verify if the experimental equipotential lines align with theoretical expectations.
   • Discuss any deviations and potential sources of error.

The laboratory experiment successfully demonstrates a method for mapping electric fields in a uniform conductor, using water as a medium.

The theoretical foundation, based on Gauss' law and Laplace's equation, provides a framework for understanding the observed phenomena. The use of low alternating voltages proves effective in approximating steady states and minimizing unwanted effects. The results obtained from mapping equipotential surfaces contribute to the understanding of electric field behavior in uniform conductors.

Two decade resistance boxes, R1 and R2, are combined to form a Rayleigh potentiometer, ensuring a constant sum of R1 + R2 = 1000Ω. • The probe, either a bare wire or a pointer, is moved through water between electrodes to identify the position where the earphones register a minimum signal. • Assuming the first and second electrodes are at a potential of V0 volts (output of the signal generator), the potential at the probe is given by V = R2 V0 / (R1 + R2).

Experimental Procedure:

1. Dry the tank and place it on graph paper to establish a coordinate system.
2. Add a small amount of water to level the tank's bottom using wedges, then uniformly fill it to a depth of about 1 cm.
3. Position two long straight electrodes parallel to each other in the middle of the tank, with a distance between them approximately two-thirds of their lengths.
4. Connect the circuit as shown in the diagram. Move the probe along a line to maintain a minimum sound in the earphones, defining this line as the equipotential.
5. Create a table of values for V, R1, and R2. Determine the best signal by varying the voltage used.
6. Set the first pair of values for R1 and R2, dip the probe into the water, and move it until the minimum sound is heard. Mark the corresponding point on the map. Repeat for other points to draw contours for the specific value of V.
7. Take other values of R1 and R2, creating a complete set of contours or equipotential lines.
8. Replace one electrode with a pin electrode, dip it into the water, and map the equipotential contour for this arrangement.
9. Repeat the process using a circular strip electrode.

After obtaining the contours for various electrode configurations, draw field lines based on background knowledge of the relationship between electric field and potential. Ensure the lines are continuous from one electrode to another while meeting the necessary conditions at the electrodes.

THE DATA COLLECTED WAS AS FOLLOWED;
(a) Using straight electrodes;

Using circular electrodes;

(C) Using pin type electrodes;

The relationship between the electric field and potential is described as follows:

(a) The difference in potential between two arbitrary points in space is a function of the electric field that pervades that space.

(b) When a charge is slowly moved infinitesimally along the -axis, the change in electric potential (db) between the initial and final positions of the charge is denoted as dip. By definition, the dip in the charge's electric energy is given by Dip = q db.

Observing the contour diagrams, it is evident that the contours are closer together near the electrodes compared to zones farther away. This phenomenon is attributed to the stronger field strength typically found near the electrodes. This observation aligns with the equation we have, indicating that the distance between the initial positions of charges influences the potential.

The shape of the electrode also plays a crucial role in determining the configuration of equipotential lines. This is directly evident in the obtained lines – straight electrodes result in straight lines, circular electrodes produce circular lines, and pin-type electrodes generate narrow contours resembling the shape of the pin end.

Determining the Direction of Force Lines using Equipotential Lines:

To ascertain the direction of force lines, we can follow the condition that field lines must be perpendicular to equipotential lines. Drawing lines perpendicular to the equipotential lines and connecting them enables us to deduce the location of the electrodes.