Figure 2.1: A sink.
To make more progress on understanding the role of electric fields in explaining electrical phenomena, we need to specify a way in which the nature of the electric field produced from a given charge distribution manifests itself in space. Hopefully you have seen from even the few examples we have done with Coulomb's Law just how important the role of symmetry can be in reducing the complexity inherent in the problem of this "specification". Since Coulomb's Law works for point charges and forces us to reduce every finite distribution of charge into points, it clearly does not make very powerful use of symmetry (the only symmetry available is that due to a point charge and the ability to reduce the task of the integration by choosing integration strategies that maximize the use of symmetry to reduce the work of evaluating the integral).
An indirect approach created by Karl Friedrich Gauss does intrinsically bring in symmetry in the formulation of the method for explaining how an electric field is generated from a charge distribution. It starts by noting that a proper description of the electric field invokes the notion of a vector field. The vector field simply specifies the direction and magnitude of vectors at any particular point in space you care to look at. Pictorially, we can imagine something like the picture below:
The arrows here depict the direction and magnitude of something that can be depicted by vectors at each point in space. This probably reminds you of something like water flowing down a sink. In fact, it could represent a number of situations in which something is coupled to an event which leads to a progression in time which tends towards zero. The vector field above could represent the velocity, at various points, in a pool of water flowing down a drain located at the origin of the coordinate system or any number of other things, e.g. the inflow of dust circling around a nascent star. This tells us of the tremendous power of vectors for description. Any kind of information can be represented in vectorial form if we make the correct associations.
Vector fields can also be used to represent electric fields. The field lines we draw are, in fact, a crude representation of a vector field. Gauss realized that a more useful quantitative representation of how things flow throughout space could be realized by being more precise about how a vector field is defined. We can start by being more precise about where the vectors should be drawn from, i.e. use a more definitive way of specifying how to draw the vector field so as to maximize the information we can get from it. We can start by being more orderly as to where we draw the vectors. If, for example, we start from a given surface, we can state what the vectors look like in defining the flow of something from a specified set of points which can be used to define an initial position. As shown in the figure below, the flow of something can be defined at the surface A and we know we have a definitive description of the flow at the position of A. Without this convention, saying "where" the flow is at any given time is somewhat ambiguous.
Although it might not appear as though this type of "defining" might be useful, it already quite powerful for talking about physical observation. For example, if you've ever noticed water flowing from a faucet set open to give a relatively small flow of water, you might have noticed that the flow narrows as the water falls away from the faucet opening.
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If we look at the flux of water, defined as the amount or mass of water flowing across an area, A1, per unit of time, then we know that the same mass of water per unit time must flow across A2, placed at a lower position. However, since gravity causes the speed of the water to increase as it falls, the cross-sectional area of the water, that is to say, the size of the stream going through A1 must be larger than the size of the stream going through A2 since the volume of water (and hence the total mass since the water has constant density) per unit time crossing A1 or A2 is given by (cross-sectional area) × (velocity). Hence, since velocity goes up, cross-sectional area must go down. Note that water reaches its terminal velocity quite quickly so that the stream reaches a constant cross-sectional area after just a small distance of travel.
To be precise about measuring flux, we should define that the flow of something through a surface is dependent on the angle between the direction of flow and the surface. Since any vector can be resolved into any set of orthogonal components, we should state that the flux is only proportional to the component of the flow vector which is perpendicular to the surface. This turns out to be such a useful case that physicists generally define area as a vector! The magnitude of this vector is what we normally think of as the size of the area and the direction is always perpendicular to the surface. Since you always have two directions in which you can be perpendicular, the convention is that the direction is chosen so as to point away from any volume which is enclosed by the surface whose area we are interested in. In other words, physics generally is interested in areas which confine (in fact, define) some volume. Pointing away from the volume distinguishes inside the volume from outside the volume. The flux is then defined by the dot product of a vector representing the flow and this vector. Consider this animation.
This shows an area in which the flux through the purple
surface goes from maximum to zero. We can state this
relationship mathematically as
| (2.1.1.1) |
The first practical application of these ideas of flux is to a mystery first explained by Newton. In his theory of Universal Gravitation, Newton speculated that his force formula for gravitational attraction between two point masses, m1 and m2, should work equally well for spherical masses m1 and m2. If we replace a spherical mass distribution by a point at the center of the spherical distribution with the same total mass, the gravitational force looks exactly the same to the universe outside the sphere.
Gauss would say that this "coincidence" works because points and spheres possess the same symmetry (spherical symmetry). Imagine a point charge, q1. The charge emits an electric field in all directions. No direction is preferred. If we replace the electric charge by a spherical distribution of charge q1 with radius R, the same condition holds. The sphere looks the same from any angle so the electric field it produces looks the same from any direction. Put any other charge further away from q1 than R and it cannot tell the difference between the point charge and the spherical charge distribution. Gauss takes advantage of this to relate his electric flux formula for the spherical charge distribution to Coulomb's Law for a point charge. Gauss states that electric flux is
| (2.1.2.2) |
and that this is proportional to the electric charge. Some comments are in order: first, note that we use an integral to define the flux. This accounts for the fact that we generally need to define the flux through an area which has to be broken into infinitesimally small pieces. The flux through each piece is evaluated and the total determined via integration. Gauss' Law is the following:
| (2.1.2.3) |
for an area which encloses (hence the circle through the
integral) a charge q, the electric flux through this area is
proportional to the charge enclosed. Assuming this to be right
for the moment, we can evaluate this for a positive spherical charge
distribution of radius R. Since the electric flux extends outward, if we
choose the surface of the sphere as the surface from which to evaluate
the flux, then E is perpendicular to the surface at each point
(since there are no preferred directions) then E · dA
is just E dA since the vectors are parallel at every point on the
spherical surface. Also, since the sphere is assumed to be uniform,
the E field has equal magnitude along this surface, hence the integral of E dA is just
EA = E(4pR2). Thus, if the proportionality constant is
called e0, then
| (2.1.2.4) |
| (2.1.2.5) |
| (2.1.2.6) |
Send comments to larryg@upenn5.hep.upenn.edu.
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