Chapter 1
Static Electricity

``Hence have arisen some new terms among us. We say B (and bodies like circumstanced) is electrized positively; A, negatively. Or rather, B is electrized plus; A, minus. And we daily in our experiments electrize plus or minus, as we think proper.''
                                                             - Benjamin Franklin

1.1.1  Electric Charge - A Property of Matter

Electrical charge, like mass, is an intrinsic property of matter. It shares many of the properties of mass in fact. The most important ones that we have discovered over the ages are that

1. Charges create and are subject to electrical forces.
2. Charges can neither be created nor destroyed, i.e. they are intrinsic to matter.

In addition, there are some things that charges do not share in common with mass:

1. Charges come in two types: designated as positive and negative.
2. Charge is conserved as stated above, but it can readily be transferred from one object to another as long as the object is not a fundamental particle (e.g. an electron).

That last caveat is important. It turns out that, ultimately, charge and mass do share the property of being intrinsic to matter, i.e. elementary particles have a definite mass and a {\it fixed} charge. Thus, the fact that we can transfer electrical charges between objects is explained by stating that this takes place by the actual exchange of charged particles. Thus, the charges involved are not created or destroyed anywhere, just transferred from one object to another by the exchange of small (essentially invisible) charged particles (which we now know are usually, but not always, electrons).

As neat as the above explanation is, it still does not explain all electrical observations you can make though. For example, two elementary particles (e.g. an electron and a proton) exert forces on each other without the exchange of anything "charged" between them. We can also see that macroscopic objects which are "neutral" can be subject to electrical forces without making any physical connection between them and any other object (this is referred to as induced charging). Hence we have to infer that "something" permeates the space between electrically charged objects to exert a force on them. We explain this "something" as being the electric field. Before describing it's properties, let's first work backwards, or operationally, i.e. let's say what the field does and let that serve as our definition of what it is.

1.1.2  Coulomb's Law

The force between charged particles was described on the basis of sensitive experiments by Charles Augustin Coulomb in a formula known as Coulomb's Law. Assume two charges, 1 and 2, are separated by a distance r₁₂. Since charges come in two types, we can have two cases: the charges are the same "sign" (i.e. both positive or both negative) or they are opposite (i.e. one is positive and the other negative). For either of these two cases, Coulomb found that the magnitude of the force between the particles was

F = k|q1||q2| r122

(1.1.0.1)

where the magnitude of the charges are expressed in units of Coulombs (C) in the SI system of units. The proportionality constant k has an experimentally determined value of 9.0 × 10⁹ N · m²/C². To complete the picture, we have to define the direction of the force and define what it acts on. Experimentally, we observe for our two cases (i.e. both charges have the same sign or the charges have opposite signs) that the forces acting look as follows:

Thus, to complete our definitions of the forces produced, we go to the vector expression of Coulomb's Law.

F21 = kq1q2r12 r123

(1.1.0.2)

This is the force on charge q₂ due to charge q₁. To get the proper direction, we need to note two things.

1. The quantities q₁ and q₂ now have signs which reflect the sign of the charges they represent, i.e. they can be either positive or negative.

2. Note that the vector direction is given by the unit vector r₁₂/r₁₂. The vector r₁₂ extends from q₁ to q₂, i.e. from the charge providing the force to the charge that the force acts on.

1.1.3  An Example of Coulomb's Law

Let's consider an application of Coulomb's Law. Since the elementary case of a single charge is already explained, let's consider three stationary charges. We will not specify the magnitudes or even the signs of the charges to emphasize that the approach to the problem of calculating the forces is independent of these actual values. The answer certainly depends on having this information, but the solution depends only on the mathematical nature of electrical forces as expressed through Coulomb's Law.

Problem:

Consider three electric charges arranged as a triangle as shown in the figure below. Find the net force acting on charge q₃.

The forces exerted by two charges, q₁ and q₂ on a third charge, q₃, must lie parallel or antiparallel to the directions given by the unit vectors ^r₁₃ and ^r₂₃, respectively.

Solution:

Charge q₁ must exert a force on q₃ which lies either along ( q₁ and q₃ have the same sign) or anti-parallel (q₁ and q₃ have opposite signs) to the vector r₁₃. The force of charge q₂ on q₃ must lie either along (q₂ and q₃ have the same sign) or anti-parallel (q₂ and q₃ have opposite signs) to the vector r₂₃. This will automatically be reflected in our answer if we adopt the convention of letting the direction we choose be from the charge causing the force toward the charge which experiences the force. The horizontal and vertical components of the forces must be

F13, x = F13cosq F13, y = -F13sinq F23, x = F23cosq F23, y = F23sinq tanq = L 2r Þ q = tan-1 L 2r (1.1.3.3)

The magnitudes of the forces are given by Coulomb's Law, so

F13 = kq1q3 r2 = kq1q3 (L2/4) + R2 F23 = kq2q3 r2 = kq2q3 (L2/4) + R2 (1.1.3.4)

Hence the net force on q₃ must be

Ftot. = (F13, x + F23, x)i +(F13, y + F23, y)j = kq3 (L2/4) + R2 [(q1 + q2)cosq·i + (q1 - q2)sinq·j] (1.1.3.5)

Again, the numerical answer depends on putting in the sign and magnitude of q₁, q₂, and q₃.

Send comments to larryg@upenn5.hep.upenn.edu.
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