Engineering Electromagnetics Lecture 5 Dr Ing Erwin Sitompul

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Engineering Electromagnetics Lecture 5 Dr. -Ing. Erwin Sitompul President University http: //zitompul. wordpress. com

Engineering Electromagnetics Lecture 5 Dr. -Ing. Erwin Sitompul President University http: //zitompul. wordpress. com 2 0 1 5 President University Erwin Sitompul EEM 5/1

Chapter 3 Electric Flux Density, Gauss’s Law, and DIvergence The Vector Operator Ñ and

Chapter 3 Electric Flux Density, Gauss’s Law, and DIvergence The Vector Operator Ñ and The Divergence Theorem n Divergence is an operation on a vector yielding a scalar, just like the dot product. n We define the del operator Ñ as a vector operator: n Then, treating the del operator as an ordinary vector, we can write: President University Erwin Sitompul EEM 5/2

Chapter 3 Electric Flux Density, Gauss’s Law, and DIvergence The Vector Operator Ñ and

Chapter 3 Electric Flux Density, Gauss’s Law, and DIvergence The Vector Operator Ñ and The Divergence Theorem n The Ñ operator does not have a specific form in other coordinate systems than rectangular coordinate system. n Nevertheless, Cylindrical Spherical President University Erwin Sitompul EEM 5/3

Chapter 3 Electric Flux Density, Gauss’s Law, and DIvergence The Vector Operator Ñ and

Chapter 3 Electric Flux Density, Gauss’s Law, and DIvergence The Vector Operator Ñ and The Divergence Theorem n We shall now give name to a theorem that we actually have obtained, the Divergence Theorem: n The first and last terms constitute the divergence theorem: “The integral of the normal component of any vector field over a closed surface is equal to the integral of the divergence of this vector field throughout the volume enclosed by the closed surface. ” President University Erwin Sitompul EEM 5/4

Chapter 3 Electric Flux Density, Gauss’s Law, and DIvergence The Vector Operator Ñ and

Chapter 3 Electric Flux Density, Gauss’s Law, and DIvergence The Vector Operator Ñ and The Divergence Theorem n Example Evaluate both sides of the divergence theorem for the field D = 2 xy ax + x 2 ay C/m 2 and the rectangular parallelepiped formed by the planes x = 0 and 1, y = 0 and 2, and z = 0 and 3. Divergence Theorem But President University Erwin Sitompul EEM 5/5

Chapter 3 Electric Flux Density, Gauss’s Law, and DIvergence The Vector Operator Ñ and

Chapter 3 Electric Flux Density, Gauss’s Law, and DIvergence The Vector Operator Ñ and The Divergence Theorem President University Erwin Sitompul EEM 5/6

Engineering Electromagnetics Chapter 4 Energy and Potential President University Erwin Sitompul EEM 5/7

Engineering Electromagnetics Chapter 4 Energy and Potential President University Erwin Sitompul EEM 5/7

Chapter 4 Energy and Potential Energy Expended in Moving a Point Charge in an

Chapter 4 Energy and Potential Energy Expended in Moving a Point Charge in an Electric Field n The electric field intensity was defined as the force on a unit test charge at that point where we wish to find the value of the electric field intensity. n To move the test charge against the electric field, we have to exert a force equal and opposite in magnitude to that exerted by the field. ► We must expend energy or do work. n To move the charge in the direction of the electric field, our energy expenditure turns out to be negative. ► We do not do the work, the field does. President University Erwin Sitompul EEM 5/8

Chapter 4 Energy and Potential Energy Expended in Moving a Point Charge in an

Chapter 4 Energy and Potential Energy Expended in Moving a Point Charge in an Electric Field n Suppose we wish to move a charge Q a distance d. L in an electric field E, the force on Q arising from the electric field is: n The component of this force in the direction d. L which we must overcome is: n The force that we apply must be equal and opposite to the force exerted by the field: n Differential work done by external source to Q is equal to: • If E and L are perpendicular, the differential work will be zero President University Erwin Sitompul EEM 5/9

Chapter 4 Energy and Potential Energy Expended in Moving a Point Charge in an

Chapter 4 Energy and Potential Energy Expended in Moving a Point Charge in an Electric Field n The work required to move the charge a finite distance is determined by integration: • The path must be specified beforehand • The charge is assumed to be at rest at both initial and final positions • W > 0 means we expend energy or do work • W < 0 means the field expends energy or do work President University Erwin Sitompul EEM 5/10

Chapter 4 Energy and Potential The Line Integral n The integral expression of previous

Chapter 4 Energy and Potential The Line Integral n The integral expression of previous equation is an example of a line integral, taking the form of integral along a prescribed path. n Without using vector notation, we should have to write: • EL: component of E along d. L n The work involved in moving a charge Q from B to A is approximately: President University Erwin Sitompul EEM 5/11

Chapter 4 Energy and Potential The Line Integral n If we assume that the

Chapter 4 Energy and Potential The Line Integral n If we assume that the electric field is uniform, n Therefore, n Since the summation can be interpreted as a line integral, the exact result for the uniform field can be obtained as: • For the case of uniform E, W does not depend on the particular path selected along which the charge is carried President University Erwin Sitompul EEM 5/12

Chapter 4 Energy and Potential The Line Integral n Example Given the nonuniform field

Chapter 4 Energy and Potential The Line Integral n Example Given the nonuniform field E = yax + xay +2 az, determine the work expended in carrying 2 C from B(1, 0, 1) to A(0. 8, 0. 6, 1) along the shorter arc of the circle x 2 + y 2 = 1, z = 1. • Differential path, rectangular coordinate • Circle equation: President University Erwin Sitompul EEM 5/13

Chapter 4 Energy and Potential The Line Integral n Example Redo the example, but

Chapter 4 Energy and Potential The Line Integral n Example Redo the example, but use the straight-line path from B to A. • Line equation: President University Erwin Sitompul EEM 5/14

Chapter 4 Energy and Potential Differential Length Rectangular Cylindrical Spherical President University Erwin Sitompul

Chapter 4 Energy and Potential Differential Length Rectangular Cylindrical Spherical President University Erwin Sitompul EEM 5/15

Chapter 4 Energy and Potential Work and Path Near an Infinite Line Charge President

Chapter 4 Energy and Potential Work and Path Near an Infinite Line Charge President University Erwin Sitompul EEM 5/16

Chapter 4 Energy and Potential Definition of Potential Difference and Potential n We already

Chapter 4 Energy and Potential Definition of Potential Difference and Potential n We already find the expression for the work W done by an external source in moving a charge Q from one point to another in an electric field E: n Potential difference V is defined as the work done by an external source in moving a unit positive charge from one point to another in an electric field: n We shall now set an agreement on the direction of movement. VAB signifies the potential difference between points A and B and is the work done in moving the unit charge from B (last named) to A (first named). President University Erwin Sitompul EEM 5/17

Chapter 4 Energy and Potential Definition of Potential Difference and Potential n Potential difference

Chapter 4 Energy and Potential Definition of Potential Difference and Potential n Potential difference is measured in joules per coulomb (J/C). However, volt (V) is defined as a more common unit. n The potential difference between points A and B is: • VAB is positive if work is done in carrying the unit positive charge from B to A n From the line-charge example, we found that the work done in taking a charge Q from ρ = a to ρ = b was: n Or, from ρ = b to ρ = a, n Thus, the potential difference between points at ρ = a to ρ = b is: President University Erwin Sitompul EEM 5/18

Chapter 4 Energy and Potential Definition of Potential Difference and Potential n For a

Chapter 4 Energy and Potential Definition of Potential Difference and Potential n For a point charge, we can find the potential difference between points A and B at radial distance r. A and r. B, choosing an origin at Q: • r. B > r. A VAB > 0, Work expended by the external source (us) • r. B < r. A VAB < 0, Work done by the electric field President University Erwin Sitompul EEM 5/19

Chapter 4 Energy and Potential Definition of Potential Difference and Potential n It is

Chapter 4 Energy and Potential Definition of Potential Difference and Potential n It is often convenient to speak of potential, or absolute potential, of a point rather than the potential difference between two points. n For this purpose, we must first specify the reference point which we consider to have zero potential. n The most universal zero reference point is “ground”, which means the potential of the surface region of the earth. n Another widely used reference point is “infinity. ” n For cylindrical coordinate, in discussing a coaxial cable, the outer conductor is selected as the zero reference for potential. n If the potential at point A is VA and that at B is VB, then: President University Erwin Sitompul EEM 5/20

Chapter 4 Energy and Potential The Potential Field of a Point Charge n In

Chapter 4 Energy and Potential The Potential Field of a Point Charge n In previous section we found an expression for the potential difference between two points located at r = r. A and r = r. B in the field of a point charge Q placed at the origin: n Any initial and final values of θ or Φ will not affect the answer. As long as the radial distance between r. A and r. B is constant, any complicated path between two points will not change the results. n This is because although d. L has r, θ, and Φ components, the electric field E only has the radial r component. President University Erwin Sitompul EEM 5/21

Chapter 4 Energy and Potential The Potential Field of a Point Charge n The

Chapter 4 Energy and Potential The Potential Field of a Point Charge n The potential difference between two points in the field of a point charge depends only on the distance of each point from the charge. n Thus, the simplest way to define a zero reference for potential in this case is to let V = 0 at infinity. n As the point r = r. B recedes to infinity, the potential at r. A becomes: President University Erwin Sitompul EEM 5/22

Chapter 4 Energy and Potential The Potential Field of a Point Charge n Generally,

Chapter 4 Energy and Potential The Potential Field of a Point Charge n Generally, n Physically, Q/4πε 0 r joules of work must be done in carrying 1 coulomb charge from infinity to any point in a distance of r meters from the charge Q. n We can also choose any point as a zero reference: with C 1 may be selected so that V = 0 at any desired value of r. President University Erwin Sitompul EEM 5/23

Chapter 4 Energy and Potential Equipotential Surface n Equipotential surface is a surface composed

Chapter 4 Energy and Potential Equipotential Surface n Equipotential surface is a surface composed of all those points having the same value of potential. n No work is involved in moving a charge around on an equipotential surface. n The equipotential surfaces in the potential field of a point charge are spheres centered at the point charge. n The equipotential surfaces in the potential field of a line charge are cylindrical surfaces axed at the line charge. n The equipotential surfaces in the potential field of a sheet of charge are surfaces parallel with the sheet of charge. President University Erwin Sitompul EEM 5/24

Chapter 4 Energy and Potential Homework 5 n D 3. 9. n D 4.

Chapter 4 Energy and Potential Homework 5 n D 3. 9. n D 4. 2. n D 4. 4. n D 4. 5. n For D 4. 4. , Replace P(1, 2, – 4) with P(1, St. ID, –BMonth). St. ID is the last two digits of your Student ID Number. Bmonth is your birth month. Example: Rudi Bravo (002201700016) was born on 3 June 2002. Rudi will do D 4. 4 with P(1, 16, – 6). n All homework problems from Hayt and Buck, 7 th Edition. n Due: Monday, 4 May 2015. President University Erwin Sitompul EEM 5/25