Maxwell s Equations 5/9/2016. EELE 3332 Electromagnetic II Chapter 9. Maxwell s Equations for static fields. Review Electrostatics and Magnetostatics

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1 Generate by Foxit PDF Creator Foxit oftware 5/9/ lectromagnetic II Chapter 9 Maxwell s quations Islamic University of Gaza lectrical ngineering Department Prof. Dr. Hala J l-khozonar Review lectrostatics an Magnetostatics lectrostatic Fiels (x,y,z) prouce by stationary charges. Magnetostatic Fiels H(x,y,z) prouce by steay (DC) currents or stationary magnet materials. In ummary: Maxwell s quations for static fiels lectrostatic Fiels an Magnetostatic Fiels o not vary with time (time invariant). They are inepenent of each other. Time epenent lectromagnetic Fiels (Dynamic fiels) prouce by time-varying currents. (x,y,z,t) H(x,y,z,t) 3 4 1

2 Generate by Foxit PDF Creator Foxit oftware 5/9/ Faraay s aw Michael Faraay ( ) Faraay s aw In 1831, Faray iscovere that a time-varying magnetic fiel woul prouce an inuce voltage (calle electromotive force or ). Accoring to Faraay s experiments, a static magnetic fiel prouces no current flow. Michael Faraay's ieas about conservation of energy le him to believe that since an electric current coul cause a magnetic fiel, a magnetic fiel shoul be able to prouce an electric current. He emonstrate this principle of inuction in Faraay iscovere that the inuce (in volts) in any close circuit is equal to the time rate of change of the magnetic flux linkage by the circuit. 6 Faraay s aw Faraay s aw N t t Change flux ue to moving permanent magnet where N is the flux linkage N is the number of turns in the circuit. is the flux through each turn. N t youtube.com/ watch?v=vwi Zjj8fo 7 8 2

3 Generate by Foxit PDF Creator Foxit oftware 5/9/216 Faraay s aw Minus ign? enz s aw Inuce MF is in irection that opposes the change in flux that cause it Transformer an motional electromotive forces For a circuit with a single turn, N 1 t In terms of an B =. l B. t where is the surface area of the circuit boune by the close path Notice that in time varying situation, both electric an magnetic fiels are present an interrelate. 11 The variation of flux with time may be cause in three ways: 1) By having a stationary loop in a time-varying B fiel. (Transformer inuction) 2) By having a time-varying loop area in static B fiel. (motional inuction) 3) By having a time-varying loop area in a time-varying B fiel. (general case, transformer an motional inuction) 12 3

4 Generate by Foxit PDF Creator Foxit oftware 5/9/216 Application: DC Generators Application: AC Generator Water turns wheel rotates magnet changes flux inuces rives current A. tationary loop in Time-arying B fiel (Transformer MF) A. tationary loop in Time-arying B fiel (Transformer MF) =. B. l t By applying toke's theorem B.. t B t B =. l. t This is one of Maxwell s equations for time-varying fiels. Note that (time-varying fiel is non-conservative)

5 Generate by Foxit PDF Creator Foxit oftware 5/9/216 B. Moving loop in static B fiel (Motional MF) Consier a conucting loop moving with uniform velocity u, the inuce in the loop is =. ( ). m l u B l B. Moving loop in static B fiel (Motional MF) It is kin of foun in electrical machines such as motors an generators. Given below is in example of c machine, where voltage is generate as the coil rotates within the magnetic fiel. This type of is calle motional or flux-cutting (ue to motion action) B. Moving loop in static B fiel (Motional MF) Another example of motional is illustrate below, where a conucting bar is moving between a pair of rails. 19 B. Moving loop in static B fiel (Motional MF) Recall that the force on a charge moving with uniform velocity u in a a magnetic fiel B is: F Qu B We efine the motional electric fiel as Fm m u B Q =. l ( u B). l m By applying toke's theorem, m.. m u B or u B =. ( ). m l u B l m m 2 5

6 Generate by Foxit PDF Creator Foxit oftware 5/9/216 B. Moving loop in static B fiel (Motional MF) = m. l ( u B). l C. Moving loop in Time arying Fiel Both Transformer an motional are present. =. l B. ( ). l u B t Notes: The integral is zero along the portion of the loop where u=. (e.g. l is taken along the ro in the shown figure. The irection of the inuce current is the same that of m or uxb. The limits of integration are selecte in the irection opposite to the irection of u x B to satisfy enz s law. (e.g. inuce current flows in the ro along a y, the integration over is along -a y ). 21 Transformer Motional B or = u B t Note that eq. t can always be applie in place of the equations in cases A, B, an C. 22 a conucting bar can slie freely over two conucting rails as shown in the figure. Calculate the inuce voltage in the bar ( a) If the bar is staione at y=8 cm an B 4 cos 1 t a mwb/m ( b) If the bar slies at a velocity u 2 a y m/s an B=4a z mwb/m ( c) If the bar slie at a velocity u 2 a m/s an 6 B=4 cos 1 t y az xample mwb/m y 6 2 z 2 ( a) we have transformer.8.6 B =-. 4(1 )(1 )sin1 t x y t y x 3 6 =4(1 )(.8)(.6)sin1 t 6 =19.2 sin 1 t ( b) This is motional = u B. l ua Ba. x a y z x xl 3 = -ubl 2(4.1 )(.6) =- 4.8 m

7 Generate by Foxit PDF Creator Foxit oftware 5/9/216 ( c) Both transformer an motional are present in this case. this problem can be solve in two ways: Metho 1: B = -. ( u B ). l t.6 y (1 )(1 )sin(1 t ) x y y x ay 4.1 cos(1 t y) a z. x ax.6 y t y t y t y t t y 6 6 t y 24cos1 t 24cos 1 8(1 )(.6)cos(1 ) 24cos 1 24cos1 4.8(1 )cos 1 24cos 1 25 Metho 2: =-, where B. t y.6 = 6 4cos(1 t ) y x y y x 6 y =-4(.6)sin(1 t y) y 6 6 = -.24sin(1 t y)+.24 sin1 t mwb But y u y ut 2t t Hence, t t t em sin(1 2 ).24sin1 mwb f t t t t =-.24(1 2)cos (1 2 ).24(1 )cos1 m cos 1 t y 24cos1 t 26 xample 9.2 The loop shown in the figure is insie a uniform magnetic fiel B=5 a x mwb/m 2. If sie DC of the loop cuts the flux lines at the frequency of 5 Hz an the loop lies in the yz-plane at the time t=, fin (a) The inuce at at t=1 ms. (b) The inuce current at t=3 ms. 27 The B fiel is time invariant, the inuce is motional = ( u B). l, where l zaz, u= a a t AD 4 cm, 2 f 1 Transform B into cylinrical corrinates: x B=B a B cosa sin a, where B =.5 u B B cos a B cos B sin u B. B cos.4(1 ).5cos.3 z a a a =.2 cos.2 cos z 6 cos m z l z z z z 28 7

8 Generate by Foxit PDF Creator Foxit oftware 5/9/216 To etermine recall that = t C ( C is the integration constant) t At t=, / 2 beacause the loop is in the yz-plane at that time, C / 2. Hence, t / 2 6 cos t / 2 =6 sin(1 t) m At t=1 ms, =6 sin(.1 ) =5.825 m (b) The current inuce is i 6 sin(1 t) ma R At t=3ms, i=6 sin(.3 ) ma=.1525 A Note that for sies AD an BC u B. l since az. a Displacement current Here we will consier Maxwell's curl equation for magnetic fiels (Ampere's aw) for time-varying conitions. For static M fiels, recall that H J (1) But the ivergence of the curl is zero: H J (2) However, the equation of continuity requires that J (3) t Thus, equations (2) an (3) are incompatible for time varying conitions!! 3 Here We must moify equation (1) to consier time-varying situation: To o this, a a term to eq. (1): J H J J (4) again, taking the ivergence, we have: H J J (5) ince J, then, J t t ince D Displacement current D D J D J = t t t (6) ubstituting eq (6) into eq (4) results in, D H J t (7) 31 Displacement current D This is Maxwell's equation ( base on H J t Ampere's circuit law) for a time varying fiel. D The term J = is known as isplacement current ensity. This t is the thir type of current ensity we have met: Conuction current ensity: J= (motion of charge in a conuctor) Convection Current Density: J= u (oesn t involve conuctors, current flows through an insulating meium, such as liqui, or vacuum). D Displacement Current Density: J = t (is a result of time-varying electric fiel). 32 8

9 Generate by Foxit PDF Creator Foxit oftware 5/9/216 Displacement current The insertion of J into Ampere s equation was one of the major contributions of Maxwell. Without the term J, the propagation of electromagnetic waves (e.g., raio or T) woul be impossible. Displacement current is the mechanism which allows electromagnetic waves to propagate in a non-conucting meium. The isplacement current is efine as: D I = J.. t H J J Note At low frequency J is usually neglecte compare with J. However, at Raio frequencies they become comparable. At the time of Maxwell, high frequency sources were not available an this equation coul not be verifie experimentally. It was years later that Hertz succeee in generating an etecting raio waves. This is one of the rare situation where mathematical argument pave the way for experimental investigation Displacement Current The shape of the surface use for Ampere s aw shouln t matter, as long as the path is the same. Apply Ampere s aw to a charging capacitor. B s = μ I C s + - +q -q Imagine a soup bowl surface, with the + plate resting near the bottom of the bowl. Apply Ampere s aw to the charging capacitor. B s = s + - +q -q The integral is zero because no current passes through the bowl. 9

10 Generate by Foxit PDF Creator Foxit oftware 5/9/216 B s = μic B s = (The equation on the right is actually incorrect, an the equation on the left is incomplete.) As the capacitor charges, the electric fiel between the plates changes. A q = C = = A = s As the charge an electric fiel change, the electric flux changes. q = = t t t + - +q -q This term has units of current. We efine the isplacement current to be I D =. t The changing electric flux through the bowl surface is equivalent to the current through the flat surface. s + - +q -q The stuff insie the gray boxes serves as your official starting equation for the isplacement current I D. B s = μ IC I D = μiencl με. encl t The generalize ( always correct ) form of Ampere s aw is then B s = μ IC I D = μiencl με. encl t Magnetic fiels are prouce by both conuction currents an time varying electric fiels. I D Why isplacement? If you put an insulator in between the plates of the capacitor, the atoms of the insulator are stretche because the electric fiel makes the protons want to go one way an the electrons the other. The process of stretching the atom involves isplacement of charge, an therefore a current. 1

11 Generate by Foxit PDF Creator Foxit oftware 5/9/216 Displacement current A typical example of isplacement current is the current through a capacitor when alternating voltage source is applie to its plates. The total current ensity is J+J. Take Amperian path as shown. Consier 2 surfaces boune by path. If surface 1 is chosen: J = H. l= J. = I = I 1 enc If surface 2 is chosen: J= Q H. l= J. = D. = I t t 2 2 Conuction to Displacement Current Ratio The conuction current ensity is given by J c The isplacement current ensity is given by J t Assume that the electric fiel is a sinusoial function of time: Then, We have Therefore cost J cos t, J sint c c max J, J J J c max max max Goo Conuctor ( I neglible) Goo Insulator ( I neglible) c o we obtain the same current for either surface. Note: In free space (or other perfect ielectric), the conuction current is zero an only isplacement current can exist. A time-varying electric fiel inuces magnetic fiel insie the capacitor xample 9.4 A parallel plate capacitor with plate area of 5 cm 2 an plate separation of 3 mm has a voltage 5 sin 1 3 t applie to its plates. Calculate the isplacement current assuming ε=2 ε. D I J., J = t D but D J = t t I C ( same as conuction current I c). t t I cos(1 t) I 147.4cos(1 t) na Maxwell s quations in Final Forms 44 11

12 Generate by Foxit PDF Creator Foxit oftware 5/9/216 The concepts of linearity, isotropy, an homogeneity of a material meium still apply for time varying fiels. lectromagnetic flow iagrams In a linear, homogeneous, an isotropic meium characterize by σ, ε, an μ, the following relations hol for time varying fiels: Consequently, the bounary conitions remain vali for time varying fiels. D P B H (H M) J u, D - D 1t 2t 1n 2n H H K, B - B 1t 2t 1n 2n 45 Figure 9.11 lectromagnetic flow iagrams showing the relationship between the potentials an vector fiels: (a) electrostatic system, (b) magnetostatic system, (c) electromagnetic system