Gausss theorem

To develop the diagrammatic representation of the field it is convenient to use a mathematical theorem due to Karl Friedrich Gauss (1777-1855), we define a quantity called electric flux through a surface as the product of the area A by the normal component of the electric field E.

When the E field is perpendicular to the surface (6 = 0°), cos 6 = 1, and the flux through the surface has its maximum value; similarly, the flux is zero when the field is parallel to the surface. (6 = 90° and cos 6 = 0). The number of field lines

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(6 is the angle between the normal n to the area A and the electric field E)

* Compare Newton's reasoning p. 126 and Figure 5.3.

passing through the surface also depends on the inclination of the surface to the field. It is easy to see that it is also proportional to cos d and therefore proportional to the flux. In the diagrammatic representation it is simple and convenient to define the electric flux (pE as the number of field lines through an area, irrespective of the angle at which the field lines cross.

Gaussian surface

Imagine that a surface (of any shape or size) encloses a volume containing one or more electric charges. The total outward flux is equal to the number of lines crossing the surface (lines out count as positive, and lines in as negative). The angle at which the lines cross at any particular point is not relevant as far as total flux is concerned.

Such an imaginary surface is known as a Gaussian surface.

Gaussian surface

Figure 10.3 Electric lines of force crossing an arbitrary closed surface.

Gaussian surface

Figure 10.3 Electric lines of force crossing an arbitrary closed surface.

If we have more than one electric charge the lines of force will be bent in different directions, depending on the positions and signs of the other charges. If there are just two charges the diagram is relatively simple to draw, as in Figure 10.4.

It is not difficult to see that, if only one or two charges are involved, and if no new lines are created or destroyed, the net number of lines crossing the surrounding surface is proportional to

Figure 10.4 Electric field due to two equal positive electric charges. The total flux crossing any surface which surrounds both charges will be twice the flux due to either charge.

Figure 10.5 Electric field due to two equal but opposite electric charges. All field lines begin on the positive charge and end on the negative charge, including those which appear to be going towards infinity. Such lines eventually come round and finish on the negative charge. Some lines never reach an arbitrary Gaussian surface which encloses both charges, but any line which exits through the surface will eventually come back. The total flux through such a surface is zero.

Figure 10.5 Electric field due to two equal but opposite electric charges. All field lines begin on the positive charge and end on the negative charge, including those which appear to be going towards infinity. Such lines eventually come round and finish on the negative charge. Some lines never reach an arbitrary Gaussian surface which encloses both charges, but any line which exits through the surface will eventually come back. The total flux through such a surface is zero.

the algebraic sum of the charges inside. It is not quite so obvious if we have many charges scattered randomly, some positive, some negative. It would not be practical to draw the field lines — they would look like a pile of spaghetti! Gauss showed that if we surround any number of charges, distributed in any fashion in space, by any closed surface, the total net flux through that surface is proportional to the net total charge enclosed. Stated as an equation, ¡EndA = q/e0, where the left-hand side expresses the

Karl Friedrich Gauss (1777-1855)

flux integrated over the surface. Note that this theorem works because the force field obeys Coulomb's law. Gauss's theorem is a very convenient way to represent diagram-matically the inverse square nature of the electric field.

In Figure 10.6 we depict a randomly shaped closed surface placed at random among a whole lot of electric charges, both positive and negative. Some of these charges are outside and some are inside the surface. The theorem still holds; the net flux outwards through the surface is proportional to the net total charge inside.

Coulomb's law expresses the electrostatic force between two charges in a way which is easy to visualise.

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Figure 10.6 Gauss's Law: The net flux through any imaginary closed surface is directly proportional to the net charge enclosed by that surface. A physical law expressed as a diagram.

Gauss's theorem expresses the same law without reference to the distribution of charges and applies to electric fields created by any number of charges distributed in space in any random manner. It expresses an intrinsic property of electric fields without reference to any particular situation and can easily be combined mathematically with other laws governing electricity and magnetism.

10.2.4 The energy in an electric field

In general, if we wish to assemble a system consisting of particles which are subject to forces, we have to do work. Where there is work there is energy. For example, a mass which is raised away from the earth's surface acquires 'potential energy'. This energy does not belong to that mass alone, but to the earth-mass system, and the same applies to any systems consisting of any number of particles under gravitational interaction.

Similarly, a system of electric charges possesses electrical potential energy. An example is an electrical capacitor such as the parallel plate capacitor illustrated in Figure 10.7.

The positive and negative charges on opposite plates of the capacitor attract one another but are held apart since they cannot cross the medium between the plates. The system resembles a stretched spring and, like a spring, possesses potential energy.

Figure 10.7 The energy of (a) an apple above the earth's surface, and (b) a charged electrical capacitor.

The energy of system (a) is due to the position of the apple relative to the earth's surface and belongs to the apple-earth system. In case (b) the energy is due to the position of the electric charges, but can we say where it is located? This question may appear to be quite trivial, but must be answered if we wish to extend the principle of conservation of energy so that energy is conserved not only over a whole system, but also locally, i.e. at every point in space.

Local conservation demands that if energy disappears from one region, it must flow across the boundaries of that region, and also that the rate of change of energy in a certain volume be equal to the total net flow of energy across the boundaries of that volume. If energy is not located at any definite point, local conservation does not make sense. We can achieve local conservation by assuming that energy is a property of the field rather than of the particles in the field. There is a certain energy density at every point in the field. This concept fits nicely into the model of both gravitational and electric fields.*

Of course, the idea of a field does not solve the mystery of 'action at a distance'. However, it enables us to make a map which indicates certain conditions at various points in space. In the case of a vector field, the relevant conditions at a given point in space depend only on the magnitude and direction of a vector at that point. We do not have to know what is happening at other places — every piece of information is contained in that vector.

* For a simple calculation of energy density for a uniform electric field E, see Appendix 10.1.

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