Abstract
The purpose of the overall experiment is to given a certain distribution of charge and map the electric field by identifying the equipotential surfaces/lines around the charges. Four templates are investigated, such as parallel strips of metallic paint, concentric rings. In the lab, we confirm that the electric field line is orthogonal to the equipotential line. We can also demonstrate that as one gets closer to an electrode, the equipotential contours approach the shape of the electrodes. For the concentric template in the lab, we use additionally analytical calculation to demonstrate. We plot the graph of n vs ln(r/9.2) and we get the slope of the graph. It shows the linear relationship between n and ln(r/9.2). R2 = 0.99184 for the plot suggests a strong linear relation. Equal potential lines are closer when electric field is strong.
Theory
In this lab, we map the electric field between two charges electrodes by finding lines of equal potential. Once these are found, we construct the field map using the knowledge that the electric field lines are orthogonal to the lines of equipotential. Ideally, we are supposed to place on the electrodes by a direct current source such as a battery. Then, the two electrodes would form a charged capacitor whose field line geometry we should like to determine; however, this implementation is difficult.
Therefore, in the lab, we allow a small current to flow from one electrode to the other by painting the electrodes on a plastics surface.
In the theory, the electric field line is orthogonal to the equipotential line. (shown in the figure below)

Electric potential is the integral of the electric field by distance r. According to the formula above, once we know where the electric field lines are, we can do the line integral and get the electric potential surface. Once we know the electric field, we can do the gradient of electric potential and get the electric field. In the points A and B, the voltage between them can be measured by connecting a galvanometer between them. Current will flow the meter if there is a potential difference between the points. If A and B are at the same potential, there is no current flow. Therefore, the two points are equal potential. By keeping point A constant, we can get a set of equal potential points by using the same method. By connecting the points, we get the resulting curve is equipotential line. In order to find another equipotential line, we can move point A to the contact between the next set of resistors and repeat. Therefore, we can get our results of a set of equipotential line. However, in the lab, the sensitivity was low by using the source of direct current. We use alternating current, since our ears are more sensitive.
When using an AC voltage source the charge on each electrode is no longer static. But, the shape of the electric field lines remains the same since the lines of equal potential do not change. Although the voltage by using AC source is changing, the equipotential field line dos not change. The electric field and potential are time varying, but at any one time, the field lines have the same geometry as the static case. Also, we use headphones instead of meter to search for equal potential points in order to find out where between two points there is no sound. If there is no current through the headphones, no sounds will be made. Therefore, it means the two points at the same potential.
By using Laplace’s equation in cylindrical coordinates, As for the concentric electrodes, this partial differential equation reduces to the solution of an ordinary differential equation, which is