Section 1: Electrostatics Charge density The most fundamental quantity of electrostatics is electric charge. Charge comes in two varieties, which are called “positive” and “negative”, because their effects tend to cancel. For example if we have +q and –q at the same point, electrically it is the same as no charge at all. Charge is conserved: it cannot be created or destroyed. The total charge of a closed system does not change. This is called global conservation of charge. It may happen, however, that charge disappear in one point in space and instantly appear in the other point due to the motion of the charge. There should be then continuous path between the points. This is called local conservation of charge which has its manifestation in the continuity equation. Charge is quantized. The fact is that electric charges come only in discrete lumps – integers multiples of the basic unit charge e. For example electron carries charge –e and proton carries charge +e. The fundamental unit of charge is so tiny that in macroscopic applications this charge quantization can be ignored. We will therefore be dealing with continuous distributions of charge treating it as a continuous fluid. The fundamental quantity which describes this charge distribution is the charge density. The charge density is the local charge per unit volume, i.e. ()r limq / V dq / dV , (1.1) V 0 where V is the volume centered at point r . The total charge in volume V is therefore Qdq()rr dV () dr3 , (1.2) VV V Since the SI unit of charge is Coulomb, the unit of charge density is C/m3. V d3r (r) r Fig.1.1 O If charge Q is localized in a point of space, say r0, this charge is called a point charge. A point charge produces an infinite charge density at the point r0. Indeed, if we take volume V centered at point r0 and tend this volume to zero, the ratio qV/ will diverge due to the constant charge qQ located in this volume independent of the magnitude of the volume. Therefore, the charge density ()r0 . At any other points rr 0 the charge density is zero, (r ) 0 . Despite the divergence of the charge density at rr 0 the integral over any volume V which contains point rr 0 is finite and is equal to the total charge in this volume: (r )dV Q . In this case the charge density is given in terms of the delta V function: 3 ()rrrQ (0 ). (1.3) In one dimension, the delta function, written (x a ) , is a mathematically improper function having the 1 properties: ()0xa , if x a (1.4) and ()1xadx . (1.5) The delta function can be given an intuitive meaning as the limit of a peaked curve such as a Lorentzian that becomes narrower and narrower, but higher and higher, in such a way that the area under the curve is always constant. 1 ()x lim . (1.6) 0 2/4 x22 From the definitions above it is evident that, for an arbitrary function f(x), f ()(xxadxfa ) (). (1.7) In three dimensions, with Cartesian coordinates, 3 ()()()()rr0000x xyyzz (1.8) is a function that vanishes everywhere except at rr 0 , and is such that 1, ifr V 33()rrdr 0 (1.9) 0 V 0, ifr0 V Note that a delta function has the dimensions of an inverse volume. For N point charges qi located at positions ri (i = 1…N) the charge density is given by N 3 ()rrrqii ( ). (1.10) i1 In addition to volume charge density we will also be dealing with surface and line charge densities. The surface charge density (r ) is given the local surface charge per unit surface, such that the charge in a small surface area da at point r is then dq (r ) da . Similarly, the line charge density (r ) is given by the local line charge per unit length dl, such that the charge in a small element of length dl at point r is then dq (r ) dl . Coulomb’s law All of electrostatics originates from the quantitative statement of Coulomb’s law concerning the force acting between charged bodies at rest with respect to each other. Coulomb, in a series of experiments, showed experimentally that the force between two small charged bodies separated in air a distance large compared to their dimensions varies directly as the magnitude of each charge, varies inversely as the square of the distance between them, is directed along the line joining the charges, and is attractive if the bodies are oppositely charged and repulsive if the bodies have the same type of charge. If the two charges located at positions r1 and r2 the force on the charge at r2 is equal to 2 qq rr F 12 21. (1.11) 4 3 0 rr21 -12 2 -1 -2 * Here we use the SI system, in which 0 is the permittivity of free space 0 = 8.85 10 C N m . Experiments show that the electrostatic force is additive so that the force on a point charge q placed at position r due to other charges qi located at positions ri (i = 1…N) is given by q N rr F q i (1.12) 4 i 3 0 i1 rr i This relation is called the superposition principle. Although the thing that eventually gets measured is a force, it is useful to introduce a concept of an electric field. The electric field is defined as the force exerted on a unit charge located at position vector r. It is a vector function of position, denoted by E. F = qE, (1.13) where F is the force, E the electric field, and q the charge. According to Coulomb the electric field at the point r due to point charges qi located at positions ri (i=1…N) is given by 1 N rr Er() q i , (1.14) 4 i 3 0 i1 rr i which is superposition of field produced by the individual point charges. In the SI system, the electric field is measured in volts per meter (V/m). Fig.1.2 The electric field can be visualized using field lines as is shown schematically in Fig.1.2 for two charges, positive and negative. Positive charges can be thought as a source of an electric field and negative charges can be though as sinks of an electric field. The magnitude of the electric field is determined by the density of the field lines. 3 d r' r- r' E(r) (r') r' r Fig.1.3 O * 2 In SI, the permittivity is defined using 00 1/c , where 0 is the permeability of free space. This constant, relevant for magnetic phenomena, will be discussed in Sec. 5. In SI it is defined to have the exact value 72 00410/ NA. 3 If there is continuous distribution of the charge it can be described by a charge density (r ) as is shown schematically in Fig.1.3. In this case the sum in Eq.(1.14) is replaced by an integral: 11rr rr Er()dq ( r ) d3 r . (1.15) 33 4400VVrr rr Similarly we can write expressions for the field produced by surface and line charge densities. For the surface charge density we find 1 rr Er() ( r ) da , (1.16) 3 4 0 S rr where the integral is taken over surface S. For the line charge density we have 1 rr Er() ( r ) dl , (1.17) 3 4 0 C rr where the integral is taken over line C. Gauss’s law The integral above is not always the most suitable form for the evaluation of electric fields. There is an important integral expression, called Gauss’s law, which is sometimes more useful and furthermore leads to a differential equation for E(r). First, we define the flux of electric field through a surface S by the integral of the normal component of the electric field over the surface, En da , where n is the normal to this surface. The flux is the measure S of the “number of field lines” passing through S. This quantity depends on the field strength and therefore controlled by the charge producing the electric field. The flux over any closed surface is a measure of the total charge inside. Indeed for charges inside the closed surface the field lines starting on positive charges should terminate on negative charges inside or pass through the surface. On the other hand, charges outside the surface will contribute nothing to the total flux, since its field lines pass in one side and out the other. This is the essence of Gauss’s law. Now we will make it quantitative. Consider a point charge q and a closed surface S, as shown in Fig. 1.4. Fig.1.4 4 Let r be the distance from the charge to a point on the surface, n be the outwardly directed unit normal to the surface at that point, da be an element of surface area. If the electric field E at the point on the surface due to the charge q makes an angle with the unit normal, then the normal component of E times the area element is: q cos Enda2 da . (1.18) 4 0 r Since E is directed along the line from the charge q to the surface element, cosda r2 d , where d is the element of solid angle subtended by da at the position of the charge. Therefore q En da d . (1.19) 4 0 If we now integrate the normal component of E over the whole surface, we find that qq/ , if is inside S Enda 0 (1.20) S 0, ifqS is outside The result for the charge outside the volume follows from the fact that the incoming flux of the electric field qqcos1 En11da2 da 1 d (1.21) 4401r 0 is canceled out by the outcoming flux of the electric field through the opposite surface qqcos2 En22da2 da 2 d , (1.22) 4402r 0 as is seen from Fig.1.5.