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CHAPTER 1
Foundations of Electrostatics
1.1 Coulomb's Law
Historically, the quantitative study of electrostatics began in 1784 with Coulomb's law, which is illustrated in Fig. 1.1. This law states that the electric force between two point charges is inversely proportional to the square of the distance between them and directly proportional to the product of the charges, with the direction of the force being along the straight line connecting the two charges. In these respects, the Coulomb force between two point charges is similar to the gravitational force between two point masses. An important difference between the two force laws is that the electric charge comes in two signs, with the force between like charges being repulsive, and that between two opposite charges being attractive.
Coulomb's law can be written
F = kqq'r/r2, (1)
giving the force on charge q due to charge q' in terms of the unit vector r that specifies the direction from q' to q. The unit vector r is a dimensionless vector defined as a vector divided by its magnitude, r = r/[absolute value of r], so Coulomb's law can also be written as
F = kqq'r/r3. (1.2)
This form of Coulomb's law is more useful in vector operations.
Because Coulomb's law is a proportionality, a constant k is included in Eqs. (1.1) and (1.2). This constant can be chosen to define the unit of electric charge. Unfortunately, several different definitions have been used for the unit of charge, and some care is required in treating the units consistently. We take time now to discuss some of the different systems.
The simplest choice, in terms of Coulomb's law, is to set k=1, and use Coulomb's law to define the unit of electric charge. This leads to the electrostatic unit (esu) of charge called the statcoulomb, which can be defined in words by the electrostatic force between two charges, each of one statcoulomb, a distance one centimeter apart, is one dyne. More simply stated, if all distances and forces are in cgs (centimeter-gram-second) units, then the charge in Coulomb's law (with k=1) is in statcoulombs.
Another choice for the definition of the unit of electric charge uses MKS (meter-kilogram-second) units in Coulomb's law with the constant k being given by k [??] 9 x 109. This defines the unit of charge called the coulomb, which could be defined in words by the electrostatic force between two charges, each of one coulomb, a distance one meter apart, is 9 x 109 Newtons.
The coulomb is somewhat more familiar than the statcoulomb because the common unit of current, the ampere, is defined as one coulomb per second. Possibly for this reason, the coulomb, and the form of Coulomb's law with k [??] 9 x 109 was adopted as part of the Systeme International (SI) system of units, and has gained almost universal usage in elementary physics textbooks. However, the study and understanding of electrostatics is considerably simpler using esu units with k=1. The SI system is also particularly awkward to use for relativistic or quantum formulations of electromagnetism. For this reason, we will consistently use esu units as part of what is called the Gaussian system of units relating the esu and emu systems of units, which will be introduced in Chapter 7. Conversions between the Gaussian system and the SI system are given in Appendix A.
In the SI system, Coulomb's law is further complicated by introducing a new quantity εo defined by
k = 1/4πεo. (1.3)
The use of the 1/4π is said to "rationalize" the units because it makes some later equations (such as Gauss's law) simpler. The constant εo is sometimes called "the permittivity of free space." This terminology is unfortunate because, in the theory of Quantum Electrodynamics (QED), which is the foundation theory for Classical Electromagnetism, a frequency dependent permittivity does arise (vacuum polarization) that has nothing to do with εo.
The unit of charge in the SI system is related to the esu unit by
1 coulomb = 3 x 109 statcoulombs. (1.4)
The number 3 x 109 relating the statcoulomb to the coulomb is related to a constant c, and the number 9 x 109 for the constant k is related to c2. The constant c has dimensions of velocity, and was originally introduced for consistency between electric and magnetic phenomena. Some years after its introduction, Maxwell showed that the constant c was, in fact, the speed of light in vacuum (see Chapter 10). Herman Minkowski then showed in Einstein's theory of Special Relativity (see Chapter 14) that space and time just referred to different directions in a completely symmetric space-time manifold. This makes c the conversion constant between the space axes and the time axis (just like the conversion between miles and feet in an American topographical map). This means that c is no longer a constant to be measured, but a specified number used to define the meter in terms of the second. This is the modern definition of the meter, which is defined so that light travels exactly 299,792,458 meters in one second. In cgs units, then
c = 2.99792458 x 1010 cm/sec, (1.5)
which is the number we will use in this text.
The defined value for c is very close to 3 x 1010 in magnitude (equal to three significant figures), and we will generally use the value 3 for conversions, with the understanding that the more accurate value could be used if greater accuracy were desired. (Whenever the numbers 3 or 9 appear in conversion equations, the more accurate value could be substituted.) The conversion numbers are also changed by various powers of 10 related to the difference between cgs and MKS units, as well as a mismatch in relating the ampere to the emu unit of current, the abampere. (Ten amperes equal one abampere.) The esu unit of charge and the emu unit of charge are related by
1 abcoulomb = c statcoulomb, (1.6)
which is the origin of Eq. (1.4) when the powers of 10 are adjusted appropriately.
As an example of the connection between the units, the magnitude of the charge on an electron is given by
e = 4.80 x 10-10 statcoulomb, (1.7)
or
e = 1.60 x 10-19 coulomb (1.8)
in SI units. It can be seen from the large negative power of 10 required in either case, that neither system of units is really appropriate for elementary particle, nuclei, atomic, or molecular physics (microscopic physics) where the electron charge is the relevant unit of charge. In Chapter 16, we will discuss other systems of units that are more appropriate for those cases.
The Coulomb's law force on each of the two charges is proportional to the product of the two charges, and each force is along their common axis. Thus Coulomb's law satisfies Newton's third law of equal and opposite forces. We could try to use the third law as a theoretical basis for the symmetrical appearance of the two charges, and the common action line of the forces, but Newton's third law is not a sturdy basis on which to build. Although it is satisfied in electrostatics, we will see in Chapter 7 that it is violated by the magnetic force between moving charges. A better principle is the conservation of linear and angular momentum. Conservation of linear momentum requires the two Coulomb forces to be equal and opposite. Conservation of angular momentum requires the two forces to be along the same action line. So we see that it is from these two conservation laws that the two forces in Coulomb's law are collinear, and the charges appear symmetrically.
The fact that the force is proportional to the first power of either charge is called linearity. A related, but logically somewhat more extended, assumption (verified by experiment) is that the Coulomb force due to two charges, located at different points, on a third is the vector sum of the two individual forces. This is called the superposition principle for the electric force. Extended to the force due to several charges, the superposition principle leads to the form
[MATHEMATICAL EXPRESSION OMITTED] (1.9)
for the force on a point charge q at r due to other point charges qn located at points rn. We can see from Eq. (1.9) that the r/r3 form of Coulomb's law is more convenient than using r/r2, because the unit vector for (r - rn) would be awkward to use.
1.2 The Electric Field
At this point, it becomes useful to find the force in two stages by introducing the concept of the electric field E, defined by
F = qE (1.10)
for the force on a point charge q due to any collection of other charges. With this definition, Coulomb's law for the electric field due to a point charge q is
E = qr/r2 (1.11)
The electric field at a point r due to a number of point charges qn, positions rn, is given by
[MATHEMATICAL EXPRESSION OMITTED] (1.12)
The effect of Eq. (1.9) can now be accomplished in two steps by first using Eq. (1.12) to find E, and then Eq. (1.10) to give the force on the point charge q located at r. Although introduced in this way as a mathematical convenience, we will see (as often happens in physics) that the electric field has important physical significance on its own, and is not merely a mathematical construct.
Equation (1.10) defines the electric field E at the point r in the presence of the charge q. However, care must be exercised if Eq. (1.10) is to be used to measure the electric field that existed at r before the charge q. was introduced. The introduction of the charge q. can polarize any nearby matter, changing the field at point r. This polarization can even produce an E field where none existed before the introduction of charge q. (This is a common phenomenom in static electricity, causing lightning as well as other effects.) For this reason, the use of a test charge to measure a pre-existing electric field is accomplished by
[MATHEMATICAL EXPRESSION OMITTED] (1.13)
where E0 is the electric field that was present before the test charge was introduced.
We have thus far limited our considerations to point charges. In principle, this is all that is needed because it is believed that all charges appear as point charges of value [+ or -]e for leptons and [+ or -]2/3e or [+ or -]1/3e for quarks (the constituents of strongly interacting matter). However, the sum in Eq. (1.12) would have of the order of 1023 terms for macroscopic objects, and be impossible to use. For this reason, the concept of a continuous charge distribution as an abstraction of a huge number of point particles is introduced. That is, in Eq. (1.12), the sum on n is first taken over a large number of the point charges qn that is still small compared with the total number of charges in a macroscopic sample. This leads to clusters of charge Δqi each having ni charges, so
[MATHEMATICAL EXPRESSION OMITTED] (1.14)
The number of charges in each cluster can be large, and yet all charges in a cluster are still at about the same point, since the total number of charges is huge for a macroscopic sample. (For instance, one million atoms are contained in a cube 10-6 cm on a side.) This means that a very large number of point charges looks like, and can be well aproximated by, a still large collection of effective point charges Δqi. Then the electric field will be given by
[MATHEMATICAL EXPRESSION OMITTED] (1.15)
In the limit that the number of charge clusters becomes infinite, (in this case, "infinity" is of the order of 1020) and the net charge in each cluster approaches zero compared to the total charge, the sum approaches an integral over charge differentials dq', and Eq. (1.15) is replaced by
[MATHEMATICAL EXPRESSION OMITTED] (1.16)
In Eq. (1.16), there are two different position vectors, which we refer to as the source vector r' and the field vector r.
The form of the differential charge element dq' depends on the type of charge distribution. The integral operator ∫ dq' becomes
[MATHEMATICAL EXPRESSION OMITTED] (1.17)
[MATHEMATICAL EXPRESSION OMITTED] (1.18)
[MATHEMATICAL EXPRESSION OMITTED] (1.19)
where dl', dA', and dτ' are differentials of length, area, and volume, respectively. Equation (1.16) can also be extended to point charges with the understanding that
[MATHEMATICAL EXPRESSION OMITTED] (1.20)
The charge distribution on which E acts can also be considered continuous, and then the force on the continuous charge distribution would be
F = ∫ dq E(r), (1.21)
with ∫ dq given as in Eqs. (1.17-1.20).
Equation (1.16) gives the static electric field for any charge distribution. The term "static," as used here, means that all time derivatives of the charge distribution are zero, or are neglected if the charges are moving. But using Eq. (1.16) is a "brute force" method that often requires complicated integration, and usually is not a practical way to find E. (With the use of modern computers, it has become more practical to sometimes just do these integrals on the computer.) There are several simple geometries for which the use of symmetry simplifies the integrals, and some examples of these are given in the problems at the end of this chapter. Aside from these simple cases, better methods are usually needed to find the electric field.
1.3 Electric Potential
The work done by the electric field in moving a charge q from a point A to a point B along a path C is given by
[MATHEMATICAL EXPRESSION OMITTED] (1.22)
where the notation [MATHEMATICAL EXPRESSION OMITTED] means that the displacement dr is always along the path defined by the curve C from A to B. The definition of a conservative force field is one for which the net work done around any closed path is zero. We now show that this is true for the E field of a point charge. This follows because the integrand of Eq. (1.22) can be written as the perfect differential d(-1/r) when E is given by Coulomb's law,
[MATHEMATICAL EXPRESSION OMITTED] (1.23)
with the final form in Eq. (1.23) being the integrand for the net work on a unit charge using Coulomb's law for E. This result can be extended to the E given by Eq. (1.15) or Eq. (1.16), because each of these are just linear sums of Coulomb's law for a single point charge. Thus
[MATHEMATICAL EXPRESSION OMITTED] (1.24)
for any static electric field integrated around any closed path, and E is said to be a conservative field. (The notation [??] indicates that the line integral is taken around a closed path.)
It follows that the work done on a unit charge by a conservative field in moving from point A to point B is independent of the path taken. This can be seen in Fig. 1.2 by picking any two points A and B on a closed path, and breaking the closed path integral into an integral from A to B along path C1, followed by an integral from B to A along path C2. Then
[MATHEMATICAL EXPRESSION OMITTED] (1.25)
so
[MATHEMATICAL EXPRESSION OMITTED] (1.26)
(Continues…)
Excerpted from "Classical Electromagnetism"
by .
Copyright © 2017 Jerrold Franklin.
Excerpted by permission of Dover Publications, Inc..
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