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The Electric Force

The nature of the force between electrically charged objects was also under study by Franklin, Joseph Priestly (whom you've heard much about already) and Henry Cavendish. The idea that forces of infinite extent should be reduced in magnitude as the inverse-square of the distance from the source of the force was already being considered from strictly logical reasoning (again the fluid model prevails here when we consider the concept of flux; you can read your supplement or wait until Physics 151 or 2 to understand more about this). Newton's theories had already indicated that gravity is reduced as the inverse-square of the distance, so it was natural for scientists to follow the same idea for electrical forces. Charles Coulomb proved the inverse-square law for electrically charged objects in the late 1700's. According to our vector notation, the force of a charged object 1 on another charged object 2 is

where and are the magnitudes of the charges (in units of Coulombs), is the distance between the charges, is the unit vector pointing along the line from 1 to 2 (see the figure below), and is one of the fundamental constants of nature. It's value is , with being the abbreviation for Coulomb. In terms of familiar items, electrons and protons have a charge magnitude of C.

There is something of a mystery as to how objects affect each other when they are not in mechanical contact. Newton wrestled mightily with the concept of ``action-at-a-distance'' and eventually concluded that it was necessary for there to be some form of ether, or intermediate medium, which made it possible for one object to transfer force to another. We now know that no ether exists. It is possible for objects to exert forces on one another without any medium to transfer the force. From our fluid notion of electrical forces, however, we still associate forces as being due to the exchange of something between the two objects. In modern terms, we associate forces as being due to which emanate from one charge (say charge 1) and permeate all of space. Any other charge (say charge 2) which is within that field interacts with the field (NOTE: not the charge 1 itself) in order to create a force. Of course, charge 2 also puts out a field of its own and this field eventually interacts with charge 1 so that Newton's Third Law is upheld.

In the case of electrical charges, we say that they act as sources (or sinks) of electric fields, . The electric field is a vector field. It creates a force by interacting with the charge as follows:

where is the second charge we referred to and is the field due to charge 1. You can actually ``see'' what the electric field due to a point charge looks like using Maple. The fieldplot command (which you get by executing the with(plots): command) will produce a vector field provided you give it the behavior of the field. For the electric field from a point charge, we know that the field has to have the characteristics determined by the Coulomb force law:

So the electric field goes down as and points radially away from the origin. In terms of , the radial distance is . The components of the field are just and , where

as you would expect. Therefore, we can make Maple plot the field for us by specifying that it goes as

Therefore, we can use the command

fieldplot([x/(x^2 + y^2)^(1.5), y/(x^2 + y^2)^(1.5)], x=-3..3,y=-3..3);

to produce a picture of the field due to a positive charge at the origin. To get the same picture for a negative charge, we just replace with and with in the fieldplot command. From these pictures, we see that positive charges are sources of the electric field. The field flows out in all directions away from the charge. Negative charges are sinks of electric field. The field points in towards the charge.