Introduction to Electrostatics

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 r £ 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, being 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 j 2 ¡ 1j j 2 ¡ 1j 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 j ¡ ij 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 j ¡ ij 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 ½(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 ½(x): (4) i ! iX=1 Z The charge density has dimension Q=L3. In terms of ½, the expression for the electric ¯eld can be written as 3 x x0 E(x) = d x0 ½(x0) ¡ (5) x x 3 Z j ¡ 0j 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 = 21= ¯ ¯ 3 (9) Za1 i=1 ¯ dx ¯ X ¯ ¯xi 6 ¯ ¯ 7 4 ¯ ¯ 5 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, ±(x) = ±(x)±(y)±(z) (10) This function has the property that 3 d x f(x)±(x x0) = f(x0) (11) Z ¡ Returning to electrostatics, we can see that the charge density of a collection of point charges can be written as a sum of delta functions: n ½(x) = q ±(x x ) (12) i ¡ i 1=Xi Thus 3 3 E(x) = d x0 ½(x0)(x x0)= x x0 ¡ j ¡ j n Z 3 3 = d x0 q ±(x0 x )(x x0)= x x0 i ¡ i ¡ j ¡ j iX=1 Z n = q (x x )= x x 3: (13) i ¡ i j ¡ ij iX=1 7 3 Gauss's Law Although Coulomb's Law is quite su±cient for ¯nding electric ¯elds and forces, the integral form in which we have expressed it is not always the most useful approach to a problem. Another integral form, called Gauss's Law, is often more useful. Let us look ¯rst at a two-dimensional version of this law. Consider a point charge q located within a closed path C. In two dimensions, the ¯eld produced by this charge is 2q(x x )= x x 2. Consider now the ¡ 0 j ¡ 0j integral around C of that component of E which is normal to the path. This normal component is E n = qcos=r, where r is the distance from ¢ the charge to the integration point on the loop. However, dl cos =r is just the in¯nitesimal angle dÁ subtended by dl at the charge. Hence we just need to integrate dÁ around the loop. 8 dφ dl q dl cos( θ ) x 0 dφ= |x - x0 | x - x 0 C θ n E q cos dlE n = dl = q dÁ (14) C ¢ C x x I I j ¡ 0j Z Since the charge is inside, the integral is 2¼; if it were outside, the integral would be 0 because over one part of the path, cos is positive and over another part it is negative with the two parts cancelling one another when the integration is completed. Thus one ¯nds that 2¼q; q inside of C dl[n E(x)] = 8 (15) C > Z ¢ <> 0; q outside of C > The three-dimensional case: works out much the same way. The ¯eld varies as 1=r2 and so one ¯nds that d2x cos =r2 is the solid angle element d subtended by the in¯nitesimal area element d2x of S at the position of the charge.

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