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4.6. DIVERGENCE 85 4.6 Divergence [m0044] In this section, we present the divergence operator, which provides a way to calculate the flux associated with a point in space. First, let us review the concept of flux. The integral of a vector field over a surface is a scalar quantity known as flux. Specifically, the flux F of a ... | Electromagnetics_Vol1_Page_100_Chunk1401 |
86 CHAPTER 4. VECTOR ANALYSIS in the Cartesian coordinate system. This is the same “∇” that appears in the definition of the gradient operator (Section 4.5) and is same operator that often arises when considering other differential operators. If we expand D in terms of its Cartesian components: D = ˆxDx + ˆyDy + ˆzDz (4... | Electromagnetics_Vol1_Page_101_Chunk1402 |
4.7. DIVERGENCE THEOREM 87 4.7 Divergence Theorem [m0046] The Divergence Theorem relates an integral over a volume to an integral over the surface bounding that volume. This is useful in a number of situations that arise in electromagnetic analysis. In this section, we derive this theorem. Consider a vector field A repr... | Electromagnetics_Vol1_Page_102_Chunk1403 |
88 CHAPTER 4. VECTOR ANALYSIS 4.8 Curl [m0048] Curl is an operation, which when applied to a vector field, quantifies the circulation of that field. The concept of circulation has several applications in electromagnetics. Two of these applications correspond to directly to Maxwell’s Equations: • The circulation of an elec... | Electromagnetics_Vol1_Page_103_Chunk1404 |
4.8. CURL 89 is the circulation as C shrinks to it’s smallest possible size. The answer in one sense is zero, since the arclength of C is zero in this limit – there is nothing to integrate over. However, if we ask instead what is the circulation per unit area in the limit, then the result should be the non-trivial valu... | Electromagnetics_Vol1_Page_104_Chunk1405 |
90 CHAPTER 4. VECTOR ANALYSIS 4.9 Stokes’ Theorem [m0051] Stokes’ Theorem relates an integral over an open surface to an integral over the curve bounding that surface. This relationship has a number of applications in electromagnetic theory. Here is the theorem: Z S (∇× A) · ds = I C A · dl (4.120) where S is the open ... | Electromagnetics_Vol1_Page_105_Chunk1406 |
4.10. THE LAPLACIAN OPERATOR 91 4.10 The Laplacian Operator [m0099] The Laplacian ∇2f of a field f(r) is the divergence of the gradient of that field: ∇2f ≜∇· (∇f) (4.121) Note that the Laplacian is essentially a definition of the second derivative with respect to the three spatial dimensions. For example, in Cartesian co... | Electromagnetics_Vol1_Page_106_Chunk1407 |
92 CHAPTER 4. VECTOR ANALYSIS Image Credits Fig. 4.1: c⃝K. Kikkeri, https://commons.wikimedia.org/wiki/File:M0006 fCartesian.svg, CC BY SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0/). Fig. 4.2: c⃝K. Kikkeri, https://commons.wikimedia.org/wiki/File:M0006 fPositionFixed.svg, CC BY SA 4.0 (https://creativecommon... | Electromagnetics_Vol1_Page_107_Chunk1408 |
Chapter 5 Electrostatics [m0116] Electrostatics is the theory of the electric field in conditions in which its behavior is independent of magnetic fields, including • The electric field associated with fixed distributions of electric charge • Capacitance (the ability of a structure to store energy in an electric field) • Th... | Electromagnetics_Vol1_Page_108_Chunk1409 |
94 CHAPTER 5. ELECTROSTATICS F q2 q1 particle Å particle Æ ÇF R c⃝K. Kikkeri CC BY SA 4.0 Figure 5.1: Coulomb’s Law describes the force per- ceived by pairs of charged particles. Subsequently, the force perceived by particle 2 is equal and opposite; i.e., equal to −F. Separately, it is known that F can be described in ... | Electromagnetics_Vol1_Page_109_Chunk1410 |
5.2. ELECTRIC FIELD DUE TO POINT CHARGES 95 5.2 Electric Field Due to Point Charges [m0103] The electric field intensity associated with a single particle bearing charge q1, located at the origin, is (Section 5.1) E(r) = ˆr q1 4πǫr2 (5.7) If this particle is instead located at some position r1, then the above expression... | Electromagnetics_Vol1_Page_110_Chunk1411 |
96 CHAPTER 5. ELECTROSTATICS in a small cell within this volume, and let ∆v be the volume of this cell. The volume charge density ρv at any point in the volume is defined as ρv ≜lim ∆v→0 ∆q ∆v = dq dv (5.16) which has units of C/m3. Since ρv is a function of position within this volume, the total charge within a volume ... | Electromagnetics_Vol1_Page_111_Chunk1412 |
5.4. ELECTRIC FIELD DUE TO A CONTINUOUS DISTRIBUTION OF CHARGE 97 shown in Section 5.6. The following example addresses a charge distribution for which Equation 5.21 is more appropriate. Example 5.2. Electric field along the axis of a ring of uniformly-distributed charge. Consider a ring of radius a in the z = 0 plane, ... | Electromagnetics_Vol1_Page_112_Chunk1413 |
98 CHAPTER 5. ELECTROSTATICS where r′ represents the varying position over S with integration. Example 5.3. Electric field along the axis of a disk of uniformly-distributed charge. Consider a circular disk of radius a in the z = 0 plane, centered on the origin, as shown in Figure 5.3. Let the charge density over this di... | Electromagnetics_Vol1_Page_113_Chunk1414 |
5.4. ELECTRIC FIELD DUE TO A CONTINUOUS DISTRIBUTION OF CHARGE 99 y x z ρρ r - rÍ a ÎÏÐ Ñ point c⃝K. Kikkeri CC BY SA 4.0 Figure 5.3: Calculating the electric field along the axis of a disk of charge. A special case of the “disk of charge” scenario considered in the preceding example is an infinite sheet of charge. The e... | Electromagnetics_Vol1_Page_114_Chunk1415 |
100 CHAPTER 5. ELECTROSTATICS 5.5 Gauss’ Law: Integral Form [m0014] Gauss’ Law is one of the four fundamental laws of classical electromagnetics, collectively known as Maxwell’s Equations. Before diving in, the reader is strongly encouraged to review Section 2.4. In that section, Gauss’ Law emerges from the interpretat... | Electromagnetics_Vol1_Page_115_Chunk1416 |
5.6. ELECTRIC FIELD DUE TO AN INFINITE LINE CHARGE USING GAUSS’ LAW 101 Additional Reading: • “Gauss’ Law” on Wikipedia. 5.6 Electric Field Due to an Infinite Line Charge using Gauss’ Law [m0149] Section 5.5 explains one application of Gauss’ Law, which is to find the electric field due to a charged particle. In this sect... | Electromagnetics_Vol1_Page_116_Chunk1417 |
102 CHAPTER 5. ELECTROSTATICS Here’s Gauss’ Law: I S D · ds = Qencl (5.54) where D is the electric flux density ǫE, S is a closed surface with outward-facing differential surface normal ds, and Qencl is the enclosed charge. The first order of business is to constrain the form of D using a symmetry argument, as follows. C... | Electromagnetics_Vol1_Page_117_Chunk1418 |
5.7. GAUSS’ LAW: DIFFERENTIAL FORM 103 z a ρ ρ l Ò Ó Ô c⃝K. Kikkeri CC BY SA 4.0 Figure 5.4: Finding the electric field of an infinite line of charge using Gauss’ Law. the line charge, and decreases in magnitude in inverse proportion to distance from the line charge. Suggestion: Check to ensure that this solution is dime... | Electromagnetics_Vol1_Page_118_Chunk1419 |
104 CHAPTER 5. ELECTROSTATICS we use this equation as a tool to find electric fields in problems involving material boundaries. There are in fact two methods to develop the desired differential equation. One method is via the definition of divergence, whereas the other is via the divergence theorem. Both methods are prese... | Electromagnetics_Vol1_Page_119_Chunk1420 |
5.8. FORCE, ENERGY, AND POTENTIAL DIFFERENCE 105 constrain possible solutions for the electric field. For that, we might also need Kirchoff’s Voltage Law; see Section 5.11. Before moving on, it is worth noting that Equation 5.66 can be solved in the special case in which there are no boundary conditions to satisfy; i.e.... | Electromagnetics_Vol1_Page_120_Chunk1421 |
106 CHAPTER 5. ELECTROSTATICS It is also worth noting that the purpose of the dot product in Equation 5.72 is to ensure that only the component of motion parallel to the direction of the electric field is included in the energy tally. This is simply because motion in any other direction cannot be due to E, and therefore... | Electromagnetics_Vol1_Page_121_Chunk1422 |
5.9. INDEPENDENCE OF PATH 107 The solution to the preceding example is simple because the direct path between the two points is parallel to the electric field. If the path between the points had been perpendicular to E, then the solution is even easier – V21 is simply zero. In all other cases, V21 is proportional to the... | Electromagnetics_Vol1_Page_122_Chunk1423 |
108 CHAPTER 5. ELECTROSTATICS A practical application of this concept is that some paths may be easier to use than others, so there may be an advantage in computing the integral in Equation 5.85 using some path other than the path actually traversed. 5.10 Kirchoff’s Voltage Law for Electrostatics: Integral Form [m0016]... | Electromagnetics_Vol1_Page_123_Chunk1424 |
5.11. KIRCHOFF’S VOLTAGE LAW FOR ELECTROSTATICS: DIFFERENTIAL FORM 109 time-varying. If the magnetic field is time-varying, then Equation 5.87 must be modified to account for the effect of the magnetic field, which is to make the right hand size potentially different from zero. The generalized version of this expression t... | Electromagnetics_Vol1_Page_124_Chunk1425 |
110 CHAPTER 5. ELECTROSTATICS Equation 5.91 is a partial differential equation. As noted above, this equation, combined with the appropriate boundary conditions, can be solved for the electric field in arbitrarily-complicated scenarios. Interestingly, it is not the only such equation available for this purpose – Gauss’ ... | Electromagnetics_Vol1_Page_125_Chunk1426 |
5.12. ELECTRIC POTENTIAL FIELD DUE TO POINT CHARGES 111 R I V1 V Õ Ö × Ø Ù ÚÛ Ü ÝÞ ßà áâ ã äå æ ç è éêë ìí îï c⃝K. Kikkeri CC BY SA 4.0 Figure 5.5: A resistor in a larger circuit, used as an example to demonstrate the concept of node voltages. this example, ground – with respect to which the potential differences at al... | Electromagnetics_Vol1_Page_126_Chunk1427 |
112 CHAPTER 5. ELECTROSTATICS Equation 5.103 gives the electric potential at a specified location due to a finite number of charged particles. The potential field due to continuous distributions of charge is addressed in Section 5.13. 5.13 Electric Potential Field due to a Continuous Distribution of Charge [m0065] The ele... | Electromagnetics_Vol1_Page_127_Chunk1428 |
5.14. ELECTRIC FIELD AS THE GRADIENT OF POTENTIAL 113 over a surface S. The surface can be divided into small patches having area ∆s. Then, the charge associated with the nth patch, located at rn, is qn = ρs(rn) ∆s (5.108) where ρs is surface charge density (units of C/m2) at rn. Substituting this expression into Equat... | Electromagnetics_Vol1_Page_128_Chunk1429 |
114 CHAPTER 5. ELECTROSTATICS Comparing the above equation to Equation 5.115, we find: E(r) = − ˆx ∂ ∂x + ˆy ∂ ∂y + ˆz ∂ ∂z V (5.120) Note that the quantity in square brackets is the gradient operator “∇” (Section 4.5). Thus, we may write E = −∇V (5.121) which is the relationship we seek. The electric field intensity... | Electromagnetics_Vol1_Page_129_Chunk1430 |
5.15. POISSON’S AND LAPLACE’S EQUATIONS 115 5.15 Poisson’s and Laplace’s Equations [m0067] The electric scalar potential field V (r), defined in Section 5.12, is useful for a number of reasons including the ability to conveniently compute potential differences (i.e., V21 = V (r2) −V (r1)) and the ability to conveniently ... | Electromagnetics_Vol1_Page_130_Chunk1431 |
116 CHAPTER 5. ELECTROSTATICS Laplace’s Equation (Equation 5.133) states that the Laplacian of the electric potential field is zero in a source-free region. Like Poisson’s Equation, Laplace’s Equation, combined with the relevant boundary conditions, can be used to solve for V (r), but only in regions that contain no cha... | Electromagnetics_Vol1_Page_131_Chunk1432 |
5.16. POTENTIAL FIELD WITHIN A PARALLEL PLATE CAPACITOR 117 d z ρ a V-+VC V- Figure 5.6: A parallel plate capacitor, as a demonstra- tion of the use of Laplace’s Equation. Since the problem has radial symmetry, ∂V/∂φ = 0. Since d ≪a, we expect the fields to be approximately constant with ρ until we get close to the edge... | Electromagnetics_Vol1_Page_132_Chunk1433 |
118 CHAPTER 5. ELECTROSTATICS 5.17 Boundary Conditions on the Electric Field Intensity (E) [m0020] In homogeneous media, electromagnetic quantities vary smoothly and continuously. At an interface between dissimilar media, however, it is possible for electromagnetic quantities to be discontinuous. These discontinuities ... | Electromagnetics_Vol1_Page_133_Chunk1434 |
5.17. BOUNDARY CONDITIONS ON THE ELECTRIC FIELD INTENSITY (E) 119 parallel sides be l. From KVL we have I C E · dl = Z A E · dl + Z B E · dl + Z C E · dl + Z D E · dl = 0 (5.145) Now, let us reduce w and l together while (1) maintaining a constant ratio w/l ≪1 and (2) keeping C centered on S. In this process, the contr... | Electromagnetics_Vol1_Page_134_Chunk1435 |
120 CHAPTER 5. ELECTROSTATICS 5.18 Boundary Conditions on the Electric Flux Density (D) [m0021] In this section, we derive boundary conditions on the electric flux density D. The considerations are quite similar to those encountered in the development of boundary conditions on the electric field intensity (E) in Section ... | Electromagnetics_Vol1_Page_135_Chunk1436 |
5.18. BOUNDARY CONDITIONS ON THE ELECTRIC FLUX DENSITY (D) 121 where the “top” and “bottom” are in Regions 1 and 2, respectively, and Qencl is the charge enclosed by S′. Now let us reduce h and a together while (1) maintaining a constant ratio h/a ≪1 and (2) keeping S′ centered on S. Because h ≪a, the area of the side ... | Electromagnetics_Vol1_Page_136_Chunk1437 |
122 CHAPTER 5. ELECTROSTATICS 5.19 Charge and Electric Field for a Perfectly Conducting Region [m0025] In this section, we consider the behavior of charge and the electric field in the vicinity of a perfect electrical conductor (PEC). First, note that the electric field – both the electric field intensity E and electric fl... | Electromagnetics_Vol1_Page_137_Chunk1438 |
5.20. DIELECTRIC MEDIA 123 − −−−−−−−− + ++ + + + + + −−− − + + + + + − − −−−− − + + + + ++ + + by C. Burks (modified) Figure 5.12: Electric field lines due to a point charge in the vicinity of PEC regions (shaded) of various shapes. is, the electric field intensities are unequal unless the permittivities in each dielectri... | Electromagnetics_Vol1_Page_138_Chunk1439 |
124 CHAPTER 5. ELECTROSTATICS 5.21 Dielectric Breakdown [m0109] The permittivity of an ideal dielectric is independent of the magnitude of an applied electric field; the material is said to be “linear.”5 However, all practical dielectrics fail in this respect with sufficiently strong electric field. Typically, the failure... | Electromagnetics_Vol1_Page_139_Chunk1440 |
5.22. CAPACITANCE 125 The capacitance of a structure depends on its ge- ometry and the permittivity of the medium sepa- rating regions of positive and negative charge. Note that capacitance does not depend on charge, which we view as either a stimulus or response from this point of view. The corresponding response or s... | Electromagnetics_Vol1_Page_140_Chunk1441 |
126 CHAPTER 5. ELECTROSTATICS Finally, solving for IT we obtain the differential form of this relationship: IT (t) = C d dtVT (t) (5.162) Additional Reading: • “Capacitance” on Wikipedia. • “Capacitor” on Wikipedia. 5.23 The Thin Parallel Plate Capacitor [m0070] Let us now determine the capacitance of a common type of ... | Electromagnetics_Vol1_Page_141_Chunk1442 |
5.23. THE THIN PARALLEL PLATE CAPACITOR 127 because the boundary conditions on the outside (outward-facing) surfaces of the plates have a significant effect in this region. In the central region of the capacitor, however, the field is not much different from the field that exists in the case of infinite plate area. In any ... | Electromagnetics_Vol1_Page_142_Chunk1443 |
128 CHAPTER 5. ELECTROSTATICS Example 5.9. Printed circuit board capacitance. Printed circuit boards commonly include a “ground plane,” which serves as the voltage datum for the board, and at least one “power plane,” which is used to distribute a DC supply voltage (See “Additional Reading” at the end of this section). ... | Electromagnetics_Vol1_Page_143_Chunk1444 |
5.24. CAPACITANCE OF A COAXIAL STRUCTURE 129 the structure has infinite length (i.e., l →∞), since then there are no fringing fields and the internal field will be utterly constant with respect to z. In the central region of a finite-length capacitor, however, the field is not much different from the field that exists in the... | Electromagnetics_Vol1_Page_144_Chunk1445 |
130 CHAPTER 5. ELECTROSTATICS To make the connection back to lumped-element transmission line model parameters (Sections 3.4 and 3.10), we simply divide by l to get the per-unit length parameter: C′ = 2πǫs ln (b/a) (5.174) Example 5.10. Capacitance of RG-59 coaxial cable. RG-59 coaxial cable consists of an inner conduc... | Electromagnetics_Vol1_Page_145_Chunk1446 |
5.25. ELECTROSTATIC ENERGY 131 where q is the charge borne by the particle and We (units of J) is the work done by moving this particle across the potential difference V . Since we are dealing with charge distributions as opposed to charged particles, it is useful to express this in terms of the contribution ∆We made t... | Electromagnetics_Vol1_Page_146_Chunk1447 |
132 CHAPTER 5. ELECTROSTATICS applications, so it is instructive to consider the implications of Equation 5.179 for this structure in particular. For the thin parallel plate capacitor, C ≈ǫA d (5.182) where A is the plate area, d is the separation between the plates, and ǫ is the permittivity of the material between th... | Electromagnetics_Vol1_Page_147_Chunk1448 |
5.25. ELECTROSTATIC ENERGY 133 Image Credits Fig. 5.1: c⃝K. Kikkeri, https://commons.wikimedia.org/wiki/File:M0102 fCoulombsLaw.svg, CC BY SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0/). Fig. 5.2: c⃝K. Kikkeri, https://commons.wikimedia.org/wiki/File:M0104 fRing.svg, CC BY SA 4.0 (https://creativecommons.org/... | Electromagnetics_Vol1_Page_148_Chunk1449 |
Chapter 6 Steady Current and Conductivity 6.1 Convection and Conduction Currents [m0110] In practice, we deal with two physical mechanisms for current: convection and conduction. The distinction between these types of current is important in electromagnetic analysis. Convection current consists of charged particles mov... | Electromagnetics_Vol1_Page_149_Chunk1450 |
6.2. CURRENT DISTRIBUTIONS 135 6.2 Current Distributions [m0101] In elementary electric circuit theory, current is the rate at which electric charge passes a particular point in a circuit. For example, 1 A is 1 C per second. In this view current is a scalar quantity, and there are only two possible directions because c... | Electromagnetics_Vol1_Page_150_Chunk1451 |
136 CHAPTER 6. STEADY CURRENT AND CONDUCTIVITY J I a z - + ds c⃝K. Kikkeri CC BY SA 4.0 Figure 6.1: Net current I and current density J in a wire of circular cross-section. circuit analysis. With these choices, we have I = Z a ρ=0 Z 2π φ=0 | Electromagnetics_Vol1_Page_151_Chunk1452 |
6.3. CONDUCTIVITY 137 Conductivity σ is expressed in units of S/m, where 1 S = 1 Ω−1. It is important to note that the current being addressed here is conduction current, and not convection current, displacement current, or some other form of current – see Section 6.2 for elaboration. Summarizing: Ohm’s Law for Electro... | Electromagnetics_Vol1_Page_152_Chunk1453 |
138 CHAPTER 6. STEADY CURRENT AND CONDUCTIVITY of good conductors; for example, metals are often modeled as perfectly-conducting equipotential volumes in order to simplify analysis. A perfect conductor is a material for which σ → ∞, E →0, and subsequently V (the electric po- tential) is constant. One final note: It is i... | Electromagnetics_Vol1_Page_153_Chunk1454 |
6.4. RESISTANCE 139 E I a z - + V z=0 z= σ c⃝K. Kikkeri CC BY SA 4.0 Figure 6.2: Analysis of the resistance of straight wire of circular cross-section. device. In this section, we address the question of how the resistance of a device can be is determined. The following example serves this purpose. Figure 6.2 shows a s... | Electromagnetics_Vol1_Page_154_Chunk1455 |
140 CHAPTER 6. STEADY CURRENT AND CONDUCTIVITY i.e., the resistance of a wire having cross-sectional area A – regardless of the shape of the cross-section, is given by the above equation. The resistance of a right cylinder of material, given by Equation 6.7, is proportional to length and inversely proportional to cross... | Electromagnetics_Vol1_Page_155_Chunk1456 |
6.5. CONDUCTANCE 141 6.5 Conductance [m0105] Conductance, like resistance (Section 6.4), is a property of devices. Specifically: Conductance G (Ω−1 or S) is the reciprocal of resistance R. Therefore, conductance depends on both the conductivity of the materials used in the device, as well as the geometry of the device. ... | Electromagnetics_Vol1_Page_156_Chunk1457 |
142 CHAPTER 6. STEADY CURRENT AND CONDUCTIVITY density that diminishes inversely with the area through which the total current flows. (It may be helpful to view J as a flux density and I as a flux, as noted in Section 6.2.) This area is simply circumference 2πρ times length l, so J = ˆρ I 2πρl (6.9) which exhibits the cor... | Electromagnetics_Vol1_Page_157_Chunk1458 |
6.6. POWER DISSIPATION IN CONDUCTING MEDIA 143 6.6 Power Dissipation in Conducting Media [m0106] The displacement of charge in response to the force exerted by an electric field constitutes a reduction in the potential energy of the system (Section 5.8). If the charge is part of a steady current, there must be an associ... | Electromagnetics_Vol1_Page_158_Chunk1459 |
144 CHAPTER 6. STEADY CURRENT AND CONDUCTIVITY This result facilitates the analysis of power dissipation in materials exhibiting loss; i.e., having finite conductivity. But what is the power dissipation in a perfectly conducting material? For such a material, σ →∞and E →0 no matter how much current is applied (Section 6... | Electromagnetics_Vol1_Page_159_Chunk1460 |
6.6. POWER DISSIPATION IN CONDUCTING MEDIA 145 Image Credits Fig. 6.1: c⃝K. Kikkeri, https://commons.wikimedia.org/wiki/File:M0101 fMatEx1.svg, CC BY SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0/). Fig. 6.2: c⃝K. Kikkeri, https://commons.wikimedia.org/wiki/File:M0071 fMatEx1.svg, CC BY SA 4.0 (https://creativ... | Electromagnetics_Vol1_Page_160_Chunk1461 |
Chapter 7 Magnetostatics [m0117] Magnetostatics is the theory of the magnetic field in conditions in which its behavior is independent of electric fields, including • The magnetic field associated with various spatial distributions of steady current • The energy associated with the magnetic field • Inductance, which is the... | Electromagnetics_Vol1_Page_161_Chunk1462 |
7.2. GAUSS’ LAW FOR MAGNETIC FIELDS: INTEGRAL FORM 147 electrostatics magnetostatics Sources static charge steady current, magnetizable material Field intensity E (V/m) H (A/m) Flux density D (C/m2) B (Wb/m2=T) Material relations D = ǫE B = µH J = σE Force on charge q F = qE F = qv × B Maxwell’s Eqs. H S D · ds = Qencl... | Electromagnetics_Vol1_Page_162_Chunk1463 |
148 CHAPTER 7. MAGNETOSTATICS c⃝Youming / K. Kikkeri CC BY SA 4.0 Figure 7.1: Gauss’ Law for Magnetostatics applied to a two-dimensional bar magnet. For the surface S = SA, every field line entering S also leaves S, so the flux through S is zero. For the surface S = SB, every field line within S remains in S, so the flux t... | Electromagnetics_Vol1_Page_163_Chunk1464 |
7.4. AMPERE’S CIRCUITAL LAW (MAGNETOSTATICS): INTEGRAL FORM 149 This is another way of saying that there is no point in space that can be considered to be the source of the magnetic field, for if it were, then the total flux through a bounding surface would be greater than zero. Said yet another way, the source of the ma... | Electromagnetics_Vol1_Page_164_Chunk1465 |
150 CHAPTER 7. MAGNETOSTATICS Note that S can be any surface that is bounded by C – not just the taut surface implied in Figure 7.2. The integral form of Ampere’s Circuital Law for magnetostatics (Equation 7.6) relates the mag- netic field along a closed path to the total cur- rent flowing through any surface bounded by ... | Electromagnetics_Vol1_Page_165_Chunk1466 |
7.5. MAGNETIC FIELD OF AN INFINITELY-LONG STRAIGHT CURRENT-BEARING WIRE 151 symmetry of the cylindrical coordinate system we choose a circular path of radius ρ in the z = 0 plane, centered at the origin. With this choice we have Iencl = I for ρ ≥a (7.8) For ρ < a, we see that Iencl < I. a steady (DC) current will be di... | Electromagnetics_Vol1_Page_166_Chunk1467 |
152 CHAPTER 7. MAGNETOSTATICS I H c⃝Jfmelero CC BY SA 4.0 (modified) Figure 7.4: Right-hand rule for the relationship be- tween the direction of current and the direction of the magnetic field. Finally, we point out another “right-hand rule” that emerges from this solution, shown in Figure 7.4 and summarized below: The m... | Electromagnetics_Vol1_Page_167_Chunk1468 |
7.6. MAGNETIC FIELD INSIDE A STRAIGHT COIL 153 Figure 7.6: Determination of the magnetic field due to DC current in a coil. c⃝Geek3 CC BY SA 3.0 Figure 7.7: Magnetic field lines inside a straight coil with closely-spaced windings. (Dotted circles repre- sent current flowing up/out from the page; crossed cir- cles represen... | Electromagnetics_Vol1_Page_168_Chunk1469 |
154 CHAPTER 7. MAGNETOSTATICS z l' ρ2 ρ1 A D C B z1 z2 cylindrical form of coil c⃝K. Kikkeri CC BY SA 4.0 Figure 7.9: Selected path of integration. length gives number of turns, and this quantity times the current through the wire is the total amount of current crossing the surface bounded by C. For the choice of C mad... | Electromagnetics_Vol1_Page_169_Chunk1470 |
7.7. MAGNETIC FIELD OF A TOROIDAL COIL 155 7.7 Magnetic Field of a Toroidal Coil [m0049] A toroid is a cylinder in which the ends are joined to form a closed loop. An example of a toroidal coil is shown in Figure 7.10. Toroidal coils are commonly used to form inductors and transformers. The principal advantage of toroi... | Electromagnetics_Vol1_Page_170_Chunk1471 |
156 CHAPTER 7. MAGNETOSTATICS x y a b I ρ C Figure 7.12: Selected path of integration. centered on the origin in the z = z0 plane, as shown in Figure 7.12. We further require C to lie entirely inside the coil, which ensures that the enclosed current includes the current of all the windings as they pass through the hole... | Electromagnetics_Vol1_Page_171_Chunk1472 |
7.8. MAGNETIC FIELD OF AN INFINITE CURRENT SHEET 157 The magnetic field everywhere outside an ideal toroidal coil is zero. Note the caveat signaled by the use of the adjective “ideal.” In a practical toroidal coil, we expect there will be some leakage of magnetic flux between the windings. In practice, this leakage can b... | Electromagnetics_Vol1_Page_172_Chunk1473 |
158 CHAPTER 7. MAGNETOSTATICS no ˆy component. When the magnetic field due to each strip is added to that of all the other strips, the ˆz component of the sum field must be zero due to symmetry. It is also clear from symmetry considerations that the magnitude of H cannot depend on x or y. Summarizing, we have determined ... | Electromagnetics_Vol1_Page_173_Chunk1474 |
7.9. AMPERE’S LAW (MAGNETOSTATICS): DIFFERENTIAL FORM 159 7.9 Ampere’s Law (Magnetostatics): Differential Form [m0118] The integral form of Amperes’ Circuital Law (ACL; Section 7.4) for magnetostatics relates the magnetic field along a closed path to the total current flowing through any surface bounded by that path. In ... | Electromagnetics_Vol1_Page_174_Chunk1475 |
160 CHAPTER 7. MAGNETOSTATICS 7.10 Boundary Conditions on the Magnetic Flux Density (B) [m0022] In homogeneous media, electromagnetic quantities vary smoothly and continuously. At an interface between dissimilar media, however, it is possible for electromagnetic quantities to be discontinuous. Continuities and disconti... | Electromagnetics_Vol1_Page_175_Chunk1476 |
7.11. BOUNDARY CONDITIONS ON THE MAGNETIC FIELD INTENSITY (H) 161 7.11 Boundary Conditions on the Magnetic Field Intensity (H) [m0023] In homogeneous media, electromagnetic quantities vary smoothly and continuously. At a boundary between dissimilar media, however, it is possible for electromagnetic quantities to be dis... | Electromagnetics_Vol1_Page_176_Chunk1477 |
162 CHAPTER 7. MAGNETOSTATICS Eliminating the common factor of ∆l and arranging terms on the left: (H2 −H1) · ˆt = Js · | Electromagnetics_Vol1_Page_177_Chunk1478 |
7.12. INDUCTANCE 163 7.12 Inductance [m0123] Current creates a magnetic field, which subsequently exerts force on other current-bearing structures. For example, the current in each winding of a coil exerts a force on every other winding of the coil. If the windings are fixed in place, then this force is unable to do work... | Electromagnetics_Vol1_Page_178_Chunk1479 |
– we require the current I to form a closed loop, we measure the magnetic flux through this loop using the sign convention of the right-hand rule, and the ratio is the inductance. Many structures consist of multiple such loops – the coil is of course one of these. In a coil, each winding | Electromagnetics_Vol1_Page_178_Chunk1480 |
164 CHAPTER 7. MAGNETOSTATICS I B ds c⃝K. Kikkeri CC BY SA 4.0 Figure 7.16: Association between a closed loop of current and the associated magnetic flux. carries the same current, and the magnetic fields of the windings add to create a magnetic field, which grows in proportion to the winding density (Section 7.6). The ma... | Electromagnetics_Vol1_Page_179_Chunk1481 |
7.13. INDUCTANCE OF A STRAIGHT COIL 165 to the inductance of a pin or lead of an electronic component. A pin or lead is not a closed loop, so the formal definition of inductance given above – ratio of magnetic flux to current – does not apply. The broader definition of inductance – the ability to store energy in a magneti... | Electromagnetics_Vol1_Page_180_Chunk1482 |
166 CHAPTER 7. MAGNETOSTATICS flux density deep inside the coil is (Section 7.6): B ≈ˆzµNI l (7.66) Is it reasonable to use this approximation here? Since inductance pertains to energy storage, the question is really what fraction of the energy is stored in a field that is well-described by this approximation, as opposed... | Electromagnetics_Vol1_Page_181_Chunk1483 |
7.14. INDUCTANCE OF A COAXIAL STRUCTURE 167 7.14 Inductance of a Coaxial Structure [m0125] Let us now determine the inductance of coaxial structure, shown in Figure 7.18. The inductance of this structure is of interest for a number of reasons – in particular, for determining the characteristic impedance of coaxial tran... | Electromagnetics_Vol1_Page_182_Chunk1484 |
168 CHAPTER 7. MAGNETOSTATICS Next, we get Φ by integrating over the magnetic flux density Φ = Z S B · ds (7.73) where S is any open surface through which all magnetic field lines within the structure must pass. Since this can be any such surface, we may as well choose the simplest one. The simplest such surface is a pla... | Electromagnetics_Vol1_Page_183_Chunk1485 |
7.15. MAGNETIC ENERGY 169 7.15 Magnetic Energy [m0127] Consider a structure exhibiting inductance; i.e., one that is able to store energy in a magnetic field in response to an applied current. This structure could be a coil, or it could be one of a variety of inductive structures that are not explicitly intended to be a... | Electromagnetics_Vol1_Page_184_Chunk1486 |
170 CHAPTER 7. MAGNETOSTATICS Substituting these expressions into Equation 7.83, we obtain Wm = 1 2 µN 2A l Hl N 2 = 1 2µH2Al (7.86) Recall that the magnetic field inside a long coil is approximately uniform. Therefore, the density of energy stored inside the coil is approximately uniform. Noting that the product A... | Electromagnetics_Vol1_Page_185_Chunk1487 |
7.16. MAGNETIC MATERIALS 171 induced in the material is aligned in the same direction as the impressed (external) magnetic field. Diamagnetic materials – including copper, gold, and silicon – do not exhibit a persistent magnetic field, and the magnetic field induced in the material is (counter to intuition!) aligned in th... | Electromagnetics_Vol1_Page_186_Chunk1488 |
172 CHAPTER 7. MAGNETOSTATICS then recent values of H must have been relatively large and positive. Similarly, If B < 0, then recent values of H must have been relatively large and negative. Furthermore, the most recent sign of H can be inferred even if the present value of H is zero. In this sense, the material “remem... | Electromagnetics_Vol1_Page_187_Chunk1489 |
7.16. MAGNETIC MATERIALS 173 Image Credits Fig. 7.1: c⃝Youming / K. Kikkeri, https://commons.wikimedia.org/wiki/File:M0018 fGLMBarMagnet (2).svg, CC BY SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0/). Fig. 7.2: c⃝K. Kikkeri, https://commons.wikimedia.org/wiki/File:M0019 fACL.svg, CC BY SA 4.0 (https://creative... | Electromagnetics_Vol1_Page_188_Chunk1490 |
Chapter 8 Time-Varying Fields 8.1 Comparison of Static and Time-Varying Electromagnetics [m0013] Students encountering time-varying electromagnetic fields for the first time have usually been exposed to electrostatics and magnetostatics already. These disciplines exhibit many similarities as summarized in Table 8.1. The ... | Electromagnetics_Vol1_Page_189_Chunk1491 |
8.2. ELECTROMAGNETIC INDUCTION 175 Electrostatics / Time-Varying Magnetostatics (Dynamic) Electric & magnetic independent possibly coupled fields are... Maxwell’s Eqns. H S D · ds = Qencl H S D · ds = Qencl (integral) H C E · dl = 0 H C E · dl = −∂ ∂t R S B · ds H S B · ds = 0 H S B · ds = 0 H C H · ds = Iencl H C H · d... | Electromagnetics_Vol1_Page_190_Chunk1492 |
176 CHAPTER 8. TIME-VARYING FIELDS rule. Since Bimp points to the left, it appears that the induced current is opposing the increase in the magnitude of the total magnetic field. • When the magnet moves away from the coil, we observe current that is negative with respect to the reference direction indicated in Figure 8.... | Electromagnetics_Vol1_Page_191_Chunk1493 |
would then be clockwise-directed so as to oppose the increase in Bimp. Therefore, the potential measured at the bottom of the right coil would be higher than the potential at the top of the right coil. The | Electromagnetics_Vol1_Page_191_Chunk1494 |
8.2. ELECTROMAGNETIC INDUCTION 177 Magnet is ... |Bimp| in coil is ... Circuit Response Bind inside coil Motionless constant V = 0, I = 0 none Moving toward coil increasing V > 0, I > 0 Pointing right Moving away from coil decreasing V < 0, I < 0 Pointing left Table 8.2: Results of the experiment associated with Figure... | Electromagnetics_Vol1_Page_192_Chunk1495 |
178 CHAPTER 8. TIME-VARYING FIELDS 8.3 Faraday’s Law [m0055] Faraday’s Law describes the generation of electric potential by a time-varying magnetic flux. This is a form of electromagnetic induction, as discussed in Section 8.2. To begin, consider the scenario shown in Figure 8.3. A single loop of wire in the presence o... | Electromagnetics_Vol1_Page_193_Chunk1496 |
8.3. FARADAY’S LAW 179 V + - S n C Figure 8.4: Relationship between the polarity of VT and orientations of C and ˆn in the planar single-loop scenario. 2. The orientation of ˆn is determined by the right hand rule, taking the direction of C to be the perimeter of the loop beginning at “−” and ending at “+” 3. B yie... | Electromagnetics_Vol1_Page_194_Chunk1497 |
180 CHAPTER 8. TIME-VARYING FIELDS referred to as transformer emf. Transformer emf is the underlying principle of operation of transformers; for more on this see Section 8.5. 2. The perimeter C – and thus the surface S over which Φ is determined – can be time-varying. For example, a wire loop might be rotated or change... | Electromagnetics_Vol1_Page_195_Chunk1498 |
8.4. INDUCTION IN A MOTIONLESS LOOP 181 To begin, remember that Faraday’s Law is a calculation of electric potential and not current. So, the approach is to first find VT , and then find the current I that flows through the gap resistance in response. The sign convention for VT is arbitrary; here, we have selected “+” and ... | Electromagnetics_Vol1_Page_196_Chunk1499 |
induced potential goes to zero when the plane of the loop is parallel to the magnetic field lines. Said another way, there is no induction unless magnetic field lines pass through the loop. | Electromagnetics_Vol1_Page_196_Chunk1500 |
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