Every charged object generates an electric field at all points in space. The properties of the electric field are entirely dependent on the source charges. The electric field from a point charge is described as  

(I.1)

Electric fields from charge distributions can be calculated either with a summation or integral over charge elements.

Gauss’s law states that the flux of the electric field  through any closed surface equals  times the total charge enclosed by the surface:

(I.2)

Gauss’s law is very useful in case of symmetry in charge distribution. We derived expressions for electric fields for a sphere, line, and sheet of charge in class by calculating flux of electric field through Gaussian surface of appropriate symmetry and applying Gauss’s law.

	(I.3)
	(I.4)
	(I.5)

 

Note that in the case of the sphere electric field at any point in space has three non-zero components. We may refer to this case as a “3D” case. In the case of the line charge the component of the electric field along the line is zero (due to symmetry) and we are left with two nonzero components. We may refer to this as a “2D” case. Finally, in the case of the charged sheet the only nonzero component of the electric field is the one perpendicular to the sheet – a “1D” case.

In this lab we will be working with “2D” case – electric field vector lies in the plane (i.e.  has zero component perpendicular to the plane). As follows from (I.4), in case of circular symmetry of charge distribution in the plane electric field magnitude at a point outside of the charge distribution is inversely proportional to the distance from the center of the charge distribution: , or

(I.6)

Here C is some constant with the dimension of electric potential (Volts). It is proportional to the charge on the circle. Electric potential difference between points  and is defined as a line integral

(I.7)

This line integral for electrostatic field  is independent of the path between points  and . The electric potential is a scalar field, that is, it has only a magnitude and no direction. To represent this field graphically, lines of equal electric potential, called equipotential lines, are drawn. The electric potential is directly related to the electric field, which is the negative gradient of the electric potential:

(I.8)

The electric field lines are always perpendicular to equipotential lines. If equipotentials are drawn in equal increments in potential, the electric field is stronger where equipotentials are more tightly packed. Using an analogy to a topographical map and gravitational force, regions with higher electric potential correspond to higher elevation (“hills”) and lower potential values correspond to lower elevation (“valleys”). A positive test charge moves from regions of higher potential toward regions with lower potential, just as a ball moves from a hill to a valley.

In this lab, we will explore these concepts by plotting equipotential lines in the space between two charged electrodes. These lines and associated electric field will be mainly determined by the shape of the charges (electrodes). Similarly, any symmetry present in the field lines will correspond to the charge symmetry.