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Chapter 2: Electrostatics - Part A What we will learn: Nature of Electric Charge Concept of Electric Field Electric Field Intensity Electric Flux Density Charge and Current Distributions Coulomb’s Law Gauss’s Law INTRODUCTION Field theory lays the foundation of electrical, electronics, telecom and computer engineering. Field theory is used in understanding the principle of CRO, Radar, satellite communication, TV reception, remote sensing, radio astronomy, microwave devices, fiber optics, electromagnetic compatibility, instrument landing systems etc. A number of phenomena, which cannot be adequately explained by circuit theory, can possibly be explained by field theory. In circuit theory, voltages and currents are the main system variables, and they are treated as scalar quantities. On the other hand, most of the variables in electromagnetic field theory are vectors having magnitude as well as direction. 1 NATURE OF ELECTROSTATIC CHARGE In general, an atom is electrically neutral (total number of positive charge of protons = total number of negative charge of electrons). However, when negatively charged electrons are removed from atoms of a body, then there will be more positive charges than the negative charges in the atoms and the body is said to be negatively charged. On the other hand, if some electrons are added to it, the number of negative charges is more than positive charges, then the body is said to be negatively charged. First Law of ELECTROSTATICS: “Like charges repel each other, opposite charges attract each other” CONCEPT OF ELECTRIC FIELD Every charged object sets up an electric field in the surrounding space. A second charge "feels" the presence of this field. The second charge is either attracted toward the initial charge or repelled from it, depending on the signs of the charges. Of course, since the second charge also has an electric field, the first charge feels its presence and is either attracted or repelled by the second charge, too. If we place two positively charged particles A and B at a distance as shown in figure below. There is a repulsive force between these two 2 particles. This force is of the action-at-a-distance type and can be felt without any intermediate medium between A and B. Let us move particle B away. Point P is the point where particle B was placed, and is now in the electric field created by particle A. When particle B is placed back at P, a force F will be exerted on the particle B by the electric field. To verify the existence of an electric field at a point, a test charge is placed at that point. If the test charge feels an electric force, then the electric field exists at that point. Thus, an electric field is said to exist at a particular point if an electric force is acting on a charged particle at that point. The magnitude and direction of the electric field vector varies from point to point in the field space. Electric field can also vary with time. In this subject we are studying on electric field produced by static electrical charges which is not changing with time. Electrostatic field - field from electric charge at rest. 3 ELECTRIC FIELD INTENSITY E A static electric charge sets up an electric field in the region of space that surrounds it. The quantity we use to measure the strength of the electric field is its intensity. The intensity of an electric field (electric field intenity) E is the force exerted by the electric field on a unit test charge. Imagine an electric charge Q located at the origin of a co-ordinate system. A test charge q at a distance r from Q experiences a force, which according to Coulomb’s law is: F= 1 4 0 Qq r̂ r2 N, where r̂ is the unit vector directing radially from Q to q. Using the above definition the intensity of the electric field surrounding the charge Q is therefore: E = F q = 1 4 0 Q N r̂ 2 C r (or V ). m This result says that an electric charge located at the origin generates an electric field (E-field) which points radially out from the origin. The strength of the electric field inversely proportional to the square (| E | 1 r2 ) of the distance from the origin. Surfaces of constant | E | form concentric spheres about the origin. If a few point charges Q1, Q2, Q3, …Qn, at distances r1, r2, r, ... rn from point P as shown in the figure below, then every point charge will be exerting force on the test charge Qt at point P, therefore the net force on Qt will be vector sum of these forces. Thus the electric 4 field intensity at point P is the vector sum of each individual electric field intensity, E E1 E 2 E3 ... En Electric field intensity E at a point is equal to the negative gradient of electric potential at that point. In other words, E is equal to the rate of fall of potential in the direction of the lines of force. E dV (V / m) dx 5 LINES OF FORCE Michael Faraday introduces the concept of lines of force or field lines to represent the electric field. These continuous lines of force emanate from a positive charge and end on a negative charge. These lines always leave or enter a conducting surface normally. The lines indicate the path that a small positive test charge would take if it were placed in the field. A line tangent to a field line indicates the direction of the electric field at that point. The arrow shows the direction of the tangential line. The lines of force do not cross among themselves. The magnitude of the electric field is represented by the number of lines of force passing perpendicularly (normally) through a surface. Thus, there are more lines of force in a strong electric field and less in a weak field. 6 ELECTRIC FLUX In SI units, one line of electric flux is defined as a tube of lines (known as Faraday tube) of force emanates from +1 C and terminates on -1C charge. Since electric flux is numerically equal to the charge, so unit of flux is measured in coulomb, which is represented by . So = Q coulombs. The number of lines of force per unit area is directly proportional to the electric field intensity at that point. They can be related by the equation below: N oE An For free space. where N = number of lines of force, An = normal surface area to the field direction at that point, o the permittivity of the space. If A1=A2=A and N1>N2, then N1/A1 > N2/A2 0E1> 0E2, and E1>E2 7 ELECTRIC FLUX DENSITY D Flux density is given by the normal flux per unit area, denotes as D. If a flux of coulombs passes normally through an area of A m2, then flux density, D A C/m2. It is related to electric field intensity by the relation, D o r E . Therefore D is a vector field similar to which can be represented by lines of force or lines of displacement (or simply "displacement"). D is a vector quantity whose direction at every point is the same as that of E but whose magnitude is o r E . One useful property of D is that its surface integral over any closed surface equals the enclosed surface charge. MAXWELL’S EQUATIONS Modern electromagnetism is based on a set of four fundamental relations known as Maxwell’s equations: . D = v, (2.la) B E=(2.lb) t . B = 0, (2.lc) D H = J+ , (2.ld) t where E and D are electric field quantities interrelated by D = E, with being the electrical permittivity of the material; 8 B and H are magnetic field quantities interrelated by B = H, with being the magnetic permeability of the material; v is the electric charge density per unit volume; and J is the current density per unit area. These equations hold in any material, including free space (vacuum), and at any spatial location (x,y,z). His equations, which he deduced from experimental observations reported by Gauss, Ampere, Faraday, and others, not only encapsulate the connection between the electric field and electric charge and between the magnetic field and electric current, but they also define the bilateral coupling between the electric and magnetic field quantities. Together with some auxiliary relations, Maxwell’s equations form the fundamental tenets of electromagnetic theory. In the static case, all charges are permanently fixed in space, or, if they move, they do so at a steady rate so that v and J are constant in time. The time derivatives (/t = 0 ) of B and D in Eqs. (2.l b) and (2.ld) are zero, and Maxwell’s equations reduce to Electrostatics . D = v, (2.2a) E = 0. (2.2b) 9 Magnetostatics . B = 0, (2. 3a) H = J. (2. 3b) The electric and magnetic fields are no longer interconnected in the static case. This allows us to study electricity and magnetism as two distinct and separate phenomena, as long as the spatial distributions of charge and current flow remain constant in time. We refer to the study of electric and magnetic phenomena under static conditions as electrostatics and magnetostatics, respectively. Electrostatics is the subject of the present chapter, and in the Chapter 3 we learn about magnetostatics. We study electrostatics not only as a prelude to the study of timevarying fields, but also because it is an important field of study in its own right. Many electronic devices and systems are based on the principles of electrostatics. They include x-ray machines, oscilloscopes, ink-jet electrostatic printers, liquid crystal displays, copying machines, capacitance keyboards, and many solid-state control devices. Electrostatics is also used in the design if medical diagnostic sensors, such as the electrocardiogram (for recording the heart’s pumping pattern) and the electroencephalogram (for recording brain activity), as well as in numerous industrial applications. 10 Charge and Current Distributions In electromagnetics, we encounter various forms of electric charge distributions, and if the charges are in motion, they constitute current distributions. Charge may be distributed by over a volume of space, across a surface, or along a line. Charge Densities We define the volume charge density v as q dq = v 0 v dv v = lim (C/m3), (2.4) where q is the charge contained in an elemental volume v. In general, v is defined at a given point in space, specified by (x, y , z) in a Cartesian coordinate system, and at a given time t. v = v (x, y, z, t). Physically, v represents the average charge per unit volume for a volume v centered at (x, y, z), with v being large enough to contain a large number of atoms and yet small enough to be regarded as a point at the macroscopic scale under consideration. The variation of v with spatial location is called its spatial distribution, or simply its distribution. The total charge contained in a given volume v is given by Q = v v dv, (C), (2.5) 11 In some cases, particularly when dealing with conductors, electric charge may be distributed across the surface of a material, in which case the relevant quantity of interest is the surface charge density s, defined as q dq = s 0 s ds s = lim (C/m2), (2.6) where q is the charge present across an elemental surface area s. Similarly, if the charge is distributed along a line, which need not be straight, we characterize the distribution in terms of the line charge density l, defined as q dq = l 0 l dl l = lim (C/m), (2.7) 12 Current Density Figure 2.2 Consider a tube of charge with volume charge density v, as shown in Figure (a) above. The charges are moving with a mean velocity u along the axis of the tube. Over a period t, the charges move a distance l = u t. The amount of charge that crosses the tube’s cross-sectional surface s in time t is therefore. q = v v = v l s = vus t, (2.8) 13 Now consider the more general case where the charges are flowing through a surface s whose surface normal n is not necessarily parallel to u, as shown in Figure (b) above. In this case, the amount of charge q flowing through s is q = v u (s cos )t q = vus t, (2.9) and the corresponding current is I = q/t = vus = Js, (2.10) J = vu (A/m2), (2.11) where J is defined as the current density in ampere per square meter. For an arbitrary surface S, the total current flowing through it is given by I= S Jds (A), (2.12) 14 There are two types of current generated by different physical mechanisms 1) Convection current – generated by actual movement of electrically charged matter. Do not obey Ohm’s law. J is called the convection current density. 2) Conduction current – where atoms of the conducting medium do not move, only the electrons in the outermost shell are moving. Obeys Ohm’s law. When the current is generated by the actual movement of electrically charged matter, it is called a convection current, and J is called the convection current density. A wind-driven charged cloud, for example, gives rise to convection current. In some cases, the charged matter constituting the convection current consists solely of charged particles, such as the electrons of an electron beam in a cathode ray tube (the picture tube of the television and computer monitors). Convection current is distinct from conduction current. In a metal wire, for example, all the positive charges and most of the negative charges cannot move; only those electrons in the outermost electronic shells of the atoms can be easily pushed from one atom to the next if the voltage is applied across the ends of the 15 wire. This movement of electrons from atom to atom gives rise conduction current. The electrons that emerge from the wire are not necessarily the same electrons that entered the wire at the other end. Because the two types of current are generated by different physical mechanisms, conduction current obeys Ohm’s law, whereas convection current does not. 16 Coulomb’s Law The electric field was introduced and defined on the basis of the results of Coulomb’s experiments on the electrical force between charged bodies. 1) An isolated charge q induces an electric field E at every point in space, and at any specific point P, E is given by E = Rˆ q 4 R 2 (2.13) where R̂ is a unit vector pointing from q to P (Figure above), R is the distance between them, and is the electrical permittivity of the medium. 17 2) In the presence of an electric field E at a given point in space, which may be due to a single charge or a distribution of many charges, the force acting on a test charge q, when the charge is placed at that point, is given by q q' F = q E = Rˆ (N). 4 R 2 (2.14) with F measured in newtons (N) and q in coulombs (C), the unit of E is (N/C). For a material with electrical permittivity , the electrical field quantities D and E are related by D=E (2.15) = r 0, (2.16) with where 0 =8.85 10-12 (1/36) 10-9 (F/m) is the electrical permittivity of free space, and r = /0 is called the relative permittivity (or dielectric constant) of the material. For most materials and under most conditions, of the material has a constant value independent of both the magnitude and direction of E. If is independent of the magnitude of E, then the material is said to be linear because D and E are related linearly, and if it is independent of direction of E, the material is said to be isotropic. 18 Electric Field due to Multiple Point Charges The expression given by Eq.(2.13) for the field E due to a single charge can be extended to find the field due to multiple point charges. We begin by considering two point charges, q1 and q2, located at position vectors R1 and R2 from the origin of a given coordinate system, as shown in Figure below. z E E2 E1 R-R1 q1 R1 P R-R2 R q2 R2 y x The electric field E is to be evaluated at a point P with position vector R. At P, the electric field E1 due to q1 alone is given by Eq. (2.13) with R, the distance between q1 and P, replaced with | R–R1| and the unit vector R̂ replaced with (R – R1)/ |R – R1|. 19 q (R R ) E1 = 1 13 4 R R 1 (V/m), (2.17a) Similarly, the electric field due to q2 alone is q2 ( R R 2 ) E2 = 4 R R 3 2 (V/m), (2.17b) The electric field obeys the principle of linear superposition. Consequently, the total electric field E at any point in space is equal to the vector sum of the electric fields induced by all the individual charges. In the present case, E = E1 + E 2 (R R1 ) (R R 2 ) 1 = [ q1 3 + q2 3 ] 4 R R1 R R2 (2.18) Generalizing the preceding result to the case of N point charges, the electric field E at position vector R caused by charges q1, q2,…,qN located at points with position vectors R1, R2, …, RN, is given by E= qi (R R i ) 1 3 4 i 1 R Ri N (V/m). (2.19) 20 Electric Field due to a Charge Distribution We now extend the results we obtained for the field caused by discrete point charges to the case of a continuous charge distribution. Consider volume v shown in Figure below. It contains a distribution of electric charge characterized by a volume charge density v, whose magnitude may vary with spatial location within v. Rq R The differential electric field at a point P due to a differential amount of charge dq = v dv contained in a differential volume dv’ is ˆ' dE =R dq 4 R' 2 (2.20) where R ' is the vector from the differential volume dv to point P. 21 R' = R - Rq and R̂' = R ' / R’ Applying the principle of linear superposition, the total electric field E can be obtained by integrating the fields contributed by all the charges making up the charge distribution. Thus, 1 E = v d E = 4 v dv' ˆ R ' v' R '2 (volume distribution).(2.21a) If the charge is distributed across a surface S with surface charge density s, then dq = s ds. Accordingly, 1 E = s d E = 4 s ds' ˆ (surface distribution) (2.21b) R ' s' 2 R' If it is distributed along a line l with a line charge density l, then dq=l dl. Accordingly 1 ˆ ' l dl ' (line distribution) E = l d E = R 4 l ' R '2 (2.21c) 22 Gauss’s Law Gauss’ law states that the total outward flux of D over any closed surface equals the total charge enclosed Q. Another words D S ds Q (2.22) E S ds Q / The total outward flux of electric field intensity over any closed surface equals the total charge enclosed by the surface divided by the permittivity . 23 The surface S referred to here need not necessarily be a physical surface, but can be any hypothetical surface, which encloses the charge. Gauss’ law is useful in helping to determine the electric field intensity of charge distribution with some symmetry conditions. The surface S is called a Gaussian Surface. Using the divergence theorem we can write D S ds = D dv = Q v Or Q= v v dv Thus D v (2.23) This is Maxwell’s first equation of electromagnetism. Sometimes this is called the differential form of Maxwell’s first equation. Gauss’s law, as given by Eq. (2.22), provides a convenient method for determining the electrostatic flux density D , thus E , when the charge distributions possesses symmetry properties that allow us to make valid assumptions about the variations of the magnitude and direction of D as a function of spatial location. Because at every point on the surface the direction of ds is the outward normal to the surface, only the normal component of D at the surface contributes to the integral in Eq.(2.22). 24 Example The integral form of Gauss’s law can be applied to determine D due to a single isolated charge q by constructing a closed, spherical, Gaussian surface S of a arbitrary radius R centered at q, as shown in figure below. From symmetry considerations, assuming that q is positive, the direction of D must be radially outward along the unit vector R̂ , and DR, the magnitude of D , must be the same at all point on the surface, defined by position vector R . 25 To successfully apply Gauss’s law, the surface S should be chosen such that, from symmetry considerations, the magnitude of D is constant and its direction is normal or tangential at every point of each subsurface of S (the surface of a cube, for example, has six subsurfaces. 26