Introduction to Electrostatics
Charles Augustin de Coulomb (1736 - 1806)
December 23, 2000
Contents
1 Coulomb’s Law 2
2 Electric Field 4
3 Gauss’s Law 7
4 Di erential Form of Gauss’s Law 9
5 An Equation for E; the Scalar Potential 10 ∇ 5.1 Conservative Potentials ...... 11
6 Poisson’s and Laplace’s Equations 13
7 Energy in the Electric Field; Capacitance; Forces 15 7.1 Conductors ...... 20 7.2 Forces on Charged Conductors ...... 22
1 8 Green’s Theorem 27 8.1 Green’s Theorem ...... 29 8.2 Applying Green’s Theorem 1 ...... 30 8.3 Applying Green’s Theorem 2 ...... 31 8.3.1 Greens Theorem with Dirichlet B.C...... 34 8.3.2 Greens Theorem with Neumann B.C...... 36
2 We shall follow the approach of Jackson, which is more or less his- torical. Thus we start with classical electrostatics, pass on to magneto- statics, add time dependence, and wind up with Maxwell’s equations. These are then expressed within the framework of special relativity. The remainder of the course is devoted to a broad range of interesting and important applications. This development may be contrasted with the more formal and el- egant approach which starts from the Maxwell equations plus special relativity and then proceeds to work out electrostatics and magneto- statics - as well as everything else - as special cases. This is the method of e.g., Landau and Lifshitz, The Classical Theory of Fields. The rst third of the course, i.e., Physics 707, deals with physics which should be familiar to everyone; what will perhaps not be familiar are the mathematical techniques and functions that will be introduced in order to solve certain kinds of problems. These are of considerable usefulness and therefore will be important to us.
1 Coulomb’s Law
By performing experiments on small charged bodies (ideally, point charges), Charles Augustin de Coulomb, working around the time of the American and French revolutions (1785), was able to empirically infer that the force between two static charged particles is proportional
3 to the inverse square of the distance between them. The following has since become known1 have as Coulomb’s Law: Given two static charges q1 and q2, there is a force acting on each of them which is:
1. Proportional to the product of the magnitudes of the charges, be- ing attractive for unlike charges and repulsive for like charges
2. Inversely proportional to the square of the distance between the charges.
3. Directed along the line between the charges.
In the form of an equation, the law states that q q x x F = k 1 2 2 1 (1) 21 x x 2 x x | 2 1| | 2 1| where charge qi is located at xi, F21 is the force on charge 2 produced by charge 1, and k is a positive constant; vectors are denoted by boldface type. In addition, the force satis es a superposition law (or principle) in that the force F on a charge q in the presence of a number of other charges qi at xi, i = 1,...,n, is simply the sum of the forces arising from each of the latter as though it were the only other charge present2. Thus, n q (x x ) F = kq i i , (2) x x 3 iX=1 | i| 1Numerous others, such as Henry Cavendish, also may legitimately have some claim to the law. 2As we shall see, the principle of superposition follows from the linearity of Maxwell’s Equations
4 given that q is at x. The constant k has units and magnitude which depend on the system of units employed. We shall adopt cgs Gaussian units. The units of mass length and time are, respectively grams (g), centimeters (cm), and seconds (sec). The unit of charge is the statcoulomb (statcoul) which is de ned by the statement that the force between two charges, each of one statcoul, one cm apart is one dyne (dyn). Then k = 1 dyn cm2/(statcoul)2. In practice one may treat k as having dimension unity while charge has dimension of M 1/2L3/2/T .
2 Electric Field
It is customary and useful to introduce the concept of the electric eld at this point. This is a vector eld, i.e., a vector function of x. It is written as E(x) and is de ned as the force that would be experienced by a charge q at x, divided by q3. Thus, for a distribution of charges qi at xi, i=1,2,...,n, n q (x x ) E(x) = i i (3) x x 3 iX=1 | i| The electric eld has the property of being independent of the ‘test’ charge q; it is a function of the charge distribution which gives rise to the force on the test charge, and, of course, of the test charge’s position. This object has dimension Q/L2 or M 1/2/L1/2T . 3This de nition is not complete. The eld has other attributes as well since it carries momentum and energy: i.e. photons
5 At this point let us introduce the charge density (x) which is the charge per unit volume at, or very close to, x. This object is needed if we would like to integrate over a source distribution instead of summing over its constituent charges. Thus a sum is replaced by an equivalent integral, n q d3x (x). (4) i → iX=1 Z The charge density has dimension Q/L3. In terms of , the expres- sion for the electric eld can be written as
3 x x0 E(x) = d x0 (x0) (5) x x 3 Z | 0| In the particular case of a distribution of discrete point charges, it is possible to recover the sum in Eq. (3) by writing the charge density in an appropriate way. To do so we introduce the Dirac delta function (x a). It is de ned by the integral f(a) = dxf(x) (x a) (6) Z where f(x) is an arbitrary continuous function of x, and the range of integration includes the point x = a. A special case is f(x) = 1 which leads to dx (x a) = 1, (7) Z demonstrating the normalization of the delta function. From the arbi- trariness of f(x), we may conclude that (x a) is zero when x is not a and su ciently singular at x = a to give the normalization property.
6 In other words, it is in essence the charge density of a point charge (in one dimension) located at x = a. Some important relations involving delta functions are as follows:
a2 d (x a) df(x) f(x) dx = (8) Za1 dx dx x=a and N a2 df(x) [f(x)]dx = 1/ (9) Za1 i=1 dx X xi In the nal expression the xi are the 0’s of f(x) between a1 and a2. A delta function in three dimensions may be built as a product of three one-dimensional delta functions. In Cartesian coordinates,