Electric Field and Gauss' Law

How can a force be exerted across what seems to be empty space? Michael Faraday suggested that because an electrically charged object, A, creates a force on another charged object, B, anywhere in space, object A must somehow change the properties of space. Object B somehow senses the change in space and experiences a force due to the properties of the space at its location. We call the changed property of space an electric field. An electric field means that the interaction is not between two distant objects, but between an object and the field at its location.

The forces exerted by electric fields can do work, transferring energy from the field to another charged object.

How can you measure an electric field? Place a small charged object at some location. If there is an electric force on it, then there is an electric field at that point. The charge on the object that is used to test the field, called the test charge, must be small enough that it doesn’t affect other charges.

The electric field $\vec{E}$ that exists at a point is the electrostatic force $\vec{F}$ experienced by a small test charge $q_0$ placed at that point divided by the charge itself:

$$\vec{E}= \dfrac{\vec{F}}{q_0}$$

The electric field is a vector, and its direction is the same as the direction of the force $\vec{F}$ on a positive test charge.

SI Unit of Electric Field: newton per coulomb (N/C)

Point Charges

A more complete understanding of the electric field concept can be gained by considering the field created by a point charge. The electric field produced by a point charge $q$ can be obtained in general terms from Coulomb's law:

$$E= \dfrac{k \left | q \right |}{r^2} $$

From that result, the electric field does not depend on the test charge $q_0$.

As in Coulomb's law, the symbol $\left | q \right |$ denotes the magnitude of $q$, without regard to whether $q$ is positive or negative. If $q$ is positive, then $\vec{E}$ is directed away from $q$. On the other hand, if $q$ is negative, then $\vec{E}$ is directed toward $q$, since a negative charge attracts a positive test charge.

The Parallel Plate Capacitor

The equation which gives the electric field of a point charge, is a very useful result. With the aid of integral calculus, this equation can be applied in a variety of situations where point charges are distributed over one or more surfaces. One such example that has considerable practical importance is the parallel plate capacitor. This device consists of two parallel metal plates, each with area $A$. A charge $+q$ is spread uniformly over one plate, while a charge $-q$ is spread uniformly over the other plate. In the region between the plates and away from the edges, the electric field points from the positive plate toward the negative plate and is perpendicular to both. It can be shown that this electric field has a magnitude of:

$$ E=\dfrac{q}{\varepsilon_0 A } = \dfrac{\sigma}{\varepsilon_0}$$

where $\varepsilon_0$ is the permittivity of free space. In this expression the Greek symbol sigma ($\sigma$) denotes the charge per unit area ($\sigma = q/A$) and is sometimes called the charge density. Except in the region near the edges, the field has the same value at all places between the plates. The field does not depend on the distance from the charges, in distinct contrast to the field created by an isolated point charge.

As we have seen, electric charges create an electric field in the space around them. It is useful to have a kind of "map" that gives the direction and strength of the field at various places. The great English physicist Michael Faraday (1791-1867) proposed an idea tlhat provides such a "map" the idea of electric field lines. Since the electric field is the electric force per unit charge, field lines are also called lines of force.

In conducting materials such as copper, electric charges move readily in response to the forces that electric fields exert. This property of conducting materials has a major effect on the electric field that can exist within and around them. Suppose that a piece of copper carries a number of excess electrons somewhere within it. Each electron would experience a force of repulsion because of the electric field of its neighbors. And, since copper is a conductor, the excess electrons move readily in response to that force. In fact, as a consequence of the $l/r^2$ dependence on distance in Coulomb's law, they rush to the surface of the copper. Once static equilibrium is established with all of the excess charge on the surface, no further movement of charge occurs. Similarly, excess positive charge also n1oves to the surface of a conductor. In general, at equilibrium under electrostatic conditions, any excess charge resides on the surface of a conductor.

Now consider the interior of the copper. The interior is electrically neutral, although there are still free electrons that can move under the influence of an electric field. The absence of a net movement of these free electrons indicates that there is no net electric field present within the conductor. In fact, the excess charges arrange themselves on the conductor surface precisely in the manner needed to make the electric field zero within the material. Thus, at equilibrium under electrostatic conditions, the electric field is zero at any point within a conducting material. This fact has some fascinating implications.

We have discusses how a point charge creates an electric field in the space around the charge. There are also many situations in which an electric field is produced by charges that are spread out over a region, rather than by a single point charge. Such an extended collection of charges is called a charge distribution. For example, the electric field within the parallel plate capacitor is produced by positive charges spread uniformly over one plate and an equal number of negative charges spread over the other plate. As we will see, Gauss' law describes the relationship between a charge distribution and the electric field it produces. This law was formulated by the German mathematician and physicist Carl Friedrich Gauss (1777-1855).

In presenting Gauss' law, it will be necessary to introduce a new idea called electric flux. The idea of flux involves botlh the electric field and the surface through which it passes. By bringing together the electric field and the surface through which it passes, we will be able to define electric flux and then present Gauss' law.