President UniversityErwin SitompulEEM 10/1 Dr.-Ing. Erwin Sitompul President University Lecture 10 Engineering Electromagnetics

President UniversityErwin SitompulEEM 10/2 Chapter 6Dielectrics and Capacitance Capacitance of a Two-Wire Line The configuration of the two-wire line consists of two parallel conducting cylinders, each of circular cross section. We shall be able to find complete information about the electric field intensity, the potential field, the surface charge density distribution, and the capacitance. This arrangement is an important type of transmission line.

President UniversityErwin SitompulEEM 10/3 Chapter 6Dielectrics and Capacitance Capacitance of a Two-Wire Line The potential field of two infinite line charges, with a positive line charge in the xz plane at x = a and a negative line at x = –a, is shown below. The potential of a single line charge with zero reference at a radius of R 0 is: The combined potential field can be written as:

President UniversityErwin SitompulEEM 10/4 Chapter 6Dielectrics and Capacitance Capacitance of a Two-Wire Line We choose R 10 = R 20, thus placing the zero reference at equal distances from each line. Expressing R 1 and R 2 in terms of x and y, To recognize the equipotential surfaces, some algebraic manipulations are necessary. Choosing an equipotential surface V = V 1, we define a dimensionless parameter K 1 as:

President UniversityErwin SitompulEEM 10/5 Chapter 6Dielectrics and Capacitance Capacitance of a Two-Wire Line After some multiplications and algebra, we obtain: The last equation shows that the V = V 1 equipotential surface is independent of z and intersects the xz plane in a circle of radius b, The center of the circle is x = h, y = 0, where:

President UniversityErwin SitompulEEM 10/6 Chapter 6Dielectrics and Capacitance Capacitance of a Two-Wire Line Let us now consider a zero-potential conducting plane located at x = 0, and a conducting cylinder of radius b and potential V 0 with its axis located a distance h from the plane. Solving the last two equations for a and K 1 in terms of b and h, The potential of the cylinder is V 0, so that: Therefore,

President UniversityErwin SitompulEEM 10/7 Capacitance of a Two-Wire Line Chapter 6Dielectrics and Capacitance Given h, b, and V 0, we may determine a, K 1, and ρ L. The capacitance between the cylinder and the plane is now available. For a length L in the z direction, Prove the equity by solving quadratic equation in e α, where cosh(α) = h/b.

President UniversityErwin SitompulEEM 10/8 Chapter 6Dielectrics and Capacitance Capacitance of a Two-Wire Line Example The black circle shows the cross section of a cylinder of 5 m radius at a potential of 100 V in free space. Its axis is 13 m away from a plane at zero potential.

President UniversityErwin SitompulEEM 10/9 Capacitance of a Two-Wire Line Chapter 6Dielectrics and Capacitance We may also identify the cylinder representing the 50 V equipotential surface by finding new values for K 1, b, and h.

President UniversityErwin SitompulEEM 10/10 Capacitance of a Two-Wire Line Chapter 6Dielectrics and Capacitance

President UniversityErwin SitompulEEM 10/11 Capacitance of a Two-Wire Line Chapter 6Dielectrics and Capacitance

President UniversityErwin SitompulEEM 10/12 Capacitance of a Two-Wire Line For the case of a conductor with b << h, then: Chapter 6Dielectrics and Capacitance

President UniversityErwin SitompulEEM 10/13 Capacitance of a Two-Wire Line For the case of a conductor with b << h, then: Chapter 6Dielectrics and Capacitance

President UniversityErwin SitompulEEM 10/14 Chapter 8 The Steady Magnetic Field Engineering Electromagnetics

President UniversityErwin SitompulEEM 10/15 The Steady Magnetic Field At this point, we shall begin our study of the magnetic field with a definition of the magnetic field itself and show how it arises from a current distribution. The relation of the steady magnetic field to its source is more complicated than is the relation of the electrostatic field to its source. Chapter 8The Steady Magnetic Field The source of the steady magnetic field may be a permanent magnet, an electric field changing linearly with time, or a direct current. Our present concern will be the magnetic field produced by a differential dc element in the free space.

President UniversityErwin SitompulEEM 10/16 Biot-Savart Law Consider a differential current element as a vanishingly small section of a current-carrying filamentary conductor. We assume a current I flowing in a differential vector length of the filament dL. Chapter 8The Steady Magnetic Field The law of Biot-Savart then states that “At any point P the magnitude of the magnetic field intensity produced by the differential element is proportional to the product of the current, the magnitude of the differential length, and the sine of the angle lying between the filament and a line connecting the filament to the point P at which the field is desired; also, the magnitude of the magnetic field intensity is inversely proportional to the square of the distance from the differential element to the point P.”

President UniversityErwin SitompulEEM 10/17 Biot-Savart Law Chapter 8The Steady Magnetic Field The Biot-Savart law may be written concisely using vector notation as The units of the magnetic field intensity H are evidently amperes per meter (A/m). Using additional subscripts to indicate the point to which each of the quantities refers,

President UniversityErwin SitompulEEM 10/18 Biot-Savart Law Chapter 8The Steady Magnetic Field It is impossible to check experimentally the law of Biot-Savart as expressed previously, because the differential current element cannot be isolated. It follows that only the integral form of the Biot-Savart law can be verified experimentally,

President UniversityErwin SitompulEEM 10/19 The Biot-Savart law may also be expressed in terms of distributed sources, such as current density J (A/m 2 ) and surface current density K (A/m). Surface current flows in a sheet of vanishingly small thickness, and the sheet’s current density J is therefore infinite. Surface current density K, however, is measured in amperes per meter width. Thus, if the surface current density is uniform, the total current I in any width b is Biot-Savart Law Chapter 8The Steady Magnetic Field where the width b is measured perpendicularly to the direction in which the current is flowing.

President UniversityErwin SitompulEEM 10/20 Thus, the differential current element I dL may be expressed in terms of surface current density K or current density J, Biot-Savart Law Chapter 8The Steady Magnetic Field and alternate forms of the Biot-Savart law can be obtained as and For a nonuniform surface current density, integration is necessary: where dN is a differential element of the path across which the current is flowing.

President UniversityErwin SitompulEEM 10/21 Biot-Savart Law Chapter 8The Steady Magnetic Field We may illustrate the application of the Biot-Savart law by considering an infinitely long straight filament. Referring to the next figure, we should recognize the symmetry of this field. As we moves along the filament, no variation of z or occur. The field point r is given by r = ρa ρ, and the source point r’ is given by r’ = z’a z. Therefore,

President UniversityErwin SitompulEEM 10/22 Biot-Savart Law Chapter 8The Steady Magnetic Field We take dL = dz’a z and the current is directed toward the increasing values of z’. Thus we have The resulting magnetic field intensity is directed to a φ direction.

President UniversityErwin SitompulEEM 10/23 Biot-Savart Law Chapter 8The Steady Magnetic Field Continuing the integration with respect to z’ only, The magnitude of the field is not a function of φ or z. It varies inversely with the distance from the filament. The direction of the magnetic-field- intensity vector is circumferential.

President UniversityErwin SitompulEEM 10/24 Biot-Savart Law Chapter 8The Steady Magnetic Field The formula to calculate the magnetic field intensity caused by a finite-length current element can be readily used: Try to derive this formula

President UniversityErwin SitompulEEM 10/25 Biot-Savart Law Example Determine H at P 2 (0.4, 0.3, 0) in the field of an 8 A filamentary current directed inward from infinity to the origin on the positive x axis, and then outward to infinity along the y axis. Chapter 8The Steady Magnetic Field What if the line goes onward to infinity along the z axis?

President UniversityErwin SitompulEEM 10/26 Homework 9 D6.6. D8.1. D8.2. Deadline: , at 08:00. Chapter 8The Steady Magnetic Field