Course:SCIE001/Physics/Reading Guides/Chapter 28

OK Let's edit this guys =D

28.1 Symmetry

Symmetry = geometric transformation (ie translatonal, rotational and reflecting) makes no physical change

Symmetry of electric field must match the symmetry of charge distribution

If a cylinder is cylindricaly symmetrical, then its charge distribution is straight out of the cylinder in all directions

What Good Is Symmetry?

Symmetry allows us to determine what shape the field has to have. Then we can use that to determine the strength of the field.

We can determine these shapes by knowing which shapes are 'incompatible' with the symmetry of the charge distribution, then we eliminate those and we would have the only shape that the field can have. (it's a rather subtle reasoning, but hopefully Douglas will tell us about some tricks!)

Three Fundamental Symmetries

Planar: field is perpendicular to the plane

Cylindrical: field is radial toward or away from the axis. Radial symmetry means Symmetrical arrangement of parts (or charges in this case) around a single main axis (Dictionary.com)

Spherical: field is radial toward or away from the centre

Refer to figure 28.6 on page 853 for the images

28.2 The Concept of Flux

This section takes us through several figures to demonstrate some understanding of how to effectively determine the charge inside a box using the electric field the merges from it, so that they can introduce the idea of electric flux (meaning 'flow' in latin).

Figure 28.7 is a nice way of showing that if we do not know what the charge is inside a box, we can figure that out by knowing the direction of the electric field that is going through it.

A closed (3-dimensional) surface through which an electric field passes is called a Gaussian surface.

Figure 28.8 shows we can draw a 2-D cross section of Gaussian surface around a charge (the shape itself MUST be 3-D, but we draw 2-D for simplicity)

Figure 28.9 shows us two different kinds of Gaussian surfaces: Cubic and Cylindrical. This Gaussian surface is most useful when its shape matches the shape of the electric field. Figure 28.9a is not the greatest surface, because no electric field is merging form top and bottom so it does not give us too much information of what might be inside. But figure 28.9b is useful, because electric field is:

1. everywhere "perpendicular" to the surface of the cylinder and

2. has the "same magnitude" at each point on the surface. These two conditions are important.

Figure 28.10 shows two Bad surfaces that does not help us much to determine what the charge inside the box might be. one is a Spherical Gaussian surface and the other one is just a Non-closed surface (remember a Gaussian surface is always "closed", so this one on figure 28.10b is non-closed, thus not Gaussian).

Please NOTE: the differences in the two features "perpendicularity" and "magnitude" of the the electric fields in the two Spherical Gaussian surfaces in Figures 28.8b and Figure 28.10a. These features are NOT the same in these two spherical Gaussian surfaces. First surface is useful, but second one is not for the two reasons mentioned above (perpendicularity and magnitude).

All of the 'above', results in two conclusions:

1. The electric field in some sense "flows" out of a closed surface that has NET positive charge in it to a closed surface that has NET negative charge(charge flows from positive to negative).

2. The electric fields pattern through the surface is simple IF the closed surface matches the symmetry of the charge distribution inside.

Our conclusion about 'FLUX' are:

1. there is an outward flux through a closed surface around a net positive charge.

2. there is an inward flux through a closed surface around a net negative charge.

3. there is no net flux through a closed surface around a region of space in which there is no net charge.

28.3 Calculating Electric Flux

The Basic Definition of Flux

The electric flux measures the amount of electric field passing through a surface of area A if the normal to the surface is tilted at an angle θ from the field.

Equation 28.2 shows the basic definition of flux using symbols:

$\Phi _{\mathbf {e} }=E_{\perp }A=EA\cos {\theta }$

where E is the magnitude of the electric field (not a vector),

A is the area of the plane through which the electric field is passing and

θ is the angle between the electric field and the area (plane)

Equation 28.3 shows the electric flux of a constant electric field in vector form:

$\Phi _{\mathbf {e} }=\mathbf {E} \cdot \mathbf {A}$

where each of E and A are vectors and vector A is perpendicular to planar surface of area A.

NOTE: please refer to figure 28.13 for images and visuals

The Electric Flux of a Nonuniform Electric Field

If the electric field is not uniform on a surface, the electric flux can still be calculated. Basically the total area A can be divided into small pieces and calculate the electric flux for each small piece and add them up.

Figure 28.14 Shows 2 different pieces ( i and j) with different Electric fields on the single surface A.

Surface Integral ( Summation of the electric fluxes through a vast number of very tiny pieces of the surface):

$\Phi _{\mathbf {e} }=\int _{s}E\,dA.$

where s stands for surface

E is an electric field vector

A is also a vector (normal to surface)

Please Note this is slightly different from integral we learn in Calculus. Here, we are finding the integral of a SURFACE not a LINE. We will learn later in the chapter how this is done, it turns out to be pretty simple and doable.

28.6 Conductors in Electrostatic Equilibrium

At the Surface of a Conductor

Recall that within a conductor the electric field is zero. This holds true even within holes of the conductor.

In a conductor, excess charge is held at the surfce. That external field is perpindicular to the surface