Chapter 4: Electric Field and Potential Distribution (Field Maps)

Illustration 1

Engineers and physicists have invented many ways to depict fields. A vector field such as the E-field is usually represented by an arrow. Traditionally, the vector field intensity (or strength) would be represented by the density of arrows per unit length along a line orthogonal to the streamline (i.e. along the equipotential). Equipotentials and streamlines are always orthogonal to each other, which is a consequence of the mathematical relation .

Now that we use powerful computers and software to make complicated electromagnetic computations, we got accustomed to representing the field magnitude either by the length of the arrow or by its color, the latter being preferable. These methods are easier to program and can represent practically any field, however complicated it may be.

A scalar field, on the other hand, such as the electric potential V, used to be and still is represented by equipotential lines or surfaces, which are the geometrical place of all points with equal potentials. For example, the equipotential surfaces of a point charge are spheres centered onto the charge itself. The equipotential surfaces of a line charge are cylinders, which are coaxial with the line charge. In the cross-section orthogonal to the charge, they appear as equipotential lines: concentric circles. Typically, the equipotentials are plotted at equal intervals, for example, every 10 V.

Still, the 2-D plot of the E-field magnitude or the potential along a line remains the most accurate illustration method.

Choose a set of images illustrating one of the following electric structures:

Ø      Charged Sphere

Ø      Cross-section of Charged Cylinder

Ø      Cross-section of Parallel-plate Line

Ø      Cross-section of Coaxial Line

Ø      Cross-section of Two-wire Line

   

Charged Sphere

A sphere of radius  cm is uniformly charged with a total charge of 1 C. Observe the potential plot in Figure 1. Note that the potential is not zero inside the sphere since this is a volume distribution of charge in a non-conducting medium. Note also that instead of using discrete equipotential surfaces, we generated a plot of continuous potential “spectrum”.

To appreciate better the field behavior inside and outside the charged sphere, have a look at Figure 2 (the E-field magnitude versus the radial distance r from the center of the sphere) and Figure 3 (the potential versus the radial distance). Notice that: (1)  inside the sphere; (2)  outside the sphere (just like the field of a point charge). Further,  outside the sphere, which is the same as the potential of a point charge.

 

 
Figures (click to enlarge)
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Charged Cylinder

A cylinder of radius  mm is uniformly charged with volume charge density  C/m3. Observe the colored field map in Figure 1, which includes both the field streamlines and the equipotential lines. Note that the field is not zero inside the cylinder since this is a volume distribution of charge in a non-conducting medium.

To appreciate better the field behavior inside and outside the charged cylinder, have a look at Figure 2 (the E-field magnitude versus the radial distance  from the center of the cylinder) and Figure 3 (the potential versus the radial distance). Notice that: (1)  inside the cylinder; (2)  outside the cylinder (just like the field of an infinite line charge). Further: (3)  inside the cylinder, where  is the potential at the surface of the cylinder, and  is the slope of the  line; (4)  outside the cylinder,  being the reference radial distance. This is in agreement with theory, which tell us that, in this case, , and .

 
Figures (click to enlarge)
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Parallel-plate Line

Observe the colored field map in Figure 1, which includes both the field streamlines and the equipotential lines.

To appreciate better the field behavior, have a look at Figure 2 (the E-field magnitude versus the vertical line intersecting the plate electrodes right in the middle) and Figure 3 (the potential versus the same vertical line). Notice that since the E-field is practically constant inside the plates, the potential is a linear function of the vertical distance, i.e., it changes linearly from 10 V to 0 V as the observation point goes from the top plate toward the bottom electrode.

 
Figures (click to enlarge)
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Coaxial Line

This coaxial line has a wire of radius  mm and a shield of radius  mm. Observe the colored field map in Figure 1, which includes both the field streamlines and the equipotential lines. Only one quarter of the cross-section is given since the structure is circularly symmetrical.

To appreciate better the field behavior, have a look at Figure 2 (the E-field magnitude versus the radial distance ) and Figure 3 (the potential versus the same distance). Notice that since the E-field in the region between the wire and the shield depends on  as , the potential dependence is  ( at the shield where ).

 
Figures (click to enlarge)
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Two-wire Line

The two-wire line consists of two parallel cylindrical conductors (wire diameter  mm) in close proximity (offset is  mm). The two wires have potentials of the same magnitude but opposite signs. Observe the colored field map in Figure 1, which includes both the field streamlines and the equipotential lines.

To appreciate better the field behavior, have a look at Figure 2 (the E-field magnitude versus the horizontal axis) and Figure 3 (the potential versus the same axis).

 
Figures (click to enlarge)
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Figure 2

Figure 3

 

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