Summary: This is Page 2 of a PLTL workshop for introductory physics, at the undergraduate sophomore level. It may also be used in Workshop Physics, or simply as supplemental material for the course.
Coulomb Law Fields - Page 2 of 6
Copyright © 2005, G. Raymond Brown, Ph.D.
Suppose there are more than one point source charges contributing to the electric field. A fairly simple example with two source charges shown in green (this group formed by positive and negative charges of equal magnitude is called a dipole) and two different field points (black dots), is diagramed in Figure 1. We assume that the source charges are fixed in space, separated by the distance
An Electric Dipole |
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Note the coordinate system that has been chosen. The origin is placed at the midpoint between the two source charges, and the vertical axis, coordinate
The total field on any field point is due to applications of Equation 1 for each of the source charges. So the total field is the sum of these individual fields. (Like the Coulomb forces, the Coulomb electric fields superpose.) So, for any group of
Remember that the meaning of the vector
In this example we explicitly apply Equation 2 to each of the chosen field points for the dipole source.
First we apply Equation 2 to our dipole and the field point at
Next we apply Equation 2 to find the field at the second field point, located at
Coulomb Vectors for the Dipole |
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Thus the geometry gives us the vectors we need to substitute into Equation 2:
Substituting the geometry results into Equation 2 gives us the statement that
For sources that are extended bodies rather than a collection of isolated points, we need a more sophisticated manner of adding up the contributions of single charges to get the total electric field at a field point. This summing procedure is provided by the integral calculus. Basically, the vector sum of Equation 2 (
The concept involved is illustrated in the diagram below. This shows an extended charged three-dimensional cylinder, with the origin at the center of the cylinder, and a field point on the axis outside the cylinder. The cylinder is shown as a wireframe, with all but one infinitesimal element of it, labeled
A 3D Cylindrical Source |
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The source shown is composed of a right circular cylinder of charge, and a field point is located on the axis of the cylinder. An infinitesimal element of the source is designated by
So we wish to apply Equation 1 to the relationship between the element of charge
Starting with the application of Equation 1 (
Note that it is the vector
(Most of the higher-level textbooks simply use
Let's represent the total charge on the cylinder by
Note that
The trick is to use a new variable that ties the local charge to the local space. It is called the volume charge density, defined as
Now we have an integration purely over spatial variables. This Equation 3 is the basic Coulomb law for the electrostatic field of an extended charged object. All of the local charge information is wrapped up in the charge density variable