Indução e Corrente em Circuitos

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do capitula 9.1 Uma espira circular condutora de raio 20 no plano 2= 0 imersa em um campo magnético B = 10 cos a Calcule a tensão induzida na espira. - dt - dt at Y 2B dr = - az 2TI = 3,77 sen (377T) = = Q V 9.2 The loop in Figure 9.20 exists in a magnetic field B = 4cos(20t)a where is directed out of the page. If the area enclosed by the circuit find the current i(t). i(t) - / 4 = 2.70 B B = V V=AI ICTI= senl Ictl= A 5a 9.3 A circuit conducting loop lies in the xy-plane as shown in Figure 9.21. The loop has a radius and resistance R = If B = 40 sin 104 ta, find the currrent. - - 50 4 = T V R R9.4 Two conducting bars slide over two stationary rails, as illustrated in the figure. If B= Wb/m2, determine the induced emf in the loop thus Vpan = - B 1.2 m 15 m/s M2 = / 5 m/s x entin logo 5.0.2.ax) ax - dy? = 9.5 - Um condutor localizado em 1,6 m move-se com velocidade 2 ax m/s em um campo magnético, B = onde ( é uma constante, Determine a tensão induzida. it = B 10. by = 200 B (BY) = B 20 - VFem = 29.9.6 A square loop of side a recedes with a uniform velocity from an infinitely long fila- ment carrying current I along a2 as shown in Figure 9.23. Assuming that p = Po at time t = 0, show that the emf induced in the loop at t > a pta Y -ax / P Pota a Y = a. Po y p ^ In P. Upon d ( = = I.U ln Pta = 9.7 A conducting rod moves with a constant velocity of 3 m/s parallel to a long straight wire carrying a current as in Figure 9.24. Calculate the emf induced in the rod and state which end is at the higher potential. ax ay I B = = Y Logo A B = 20 cm 40 cm 60.70-2 = FIGURE 9.24 For Problem 9.7. . de A = - 9,881 V A, = & A9.8 A conducting rod has one end grounded at the origin, while the other end is free to move in = The rod rotates at 30 rad/s in a static magnetic field B = 60a, If the rod is 8 cm long, find the voltage induced in the rod. de = dpap ap = = S 2 = = 9.9 A rectangular coil has a cross-sectional area of 30 and 50 turns. If the coil rotates at 60 rad/s in a magnetic field of 0.2 such that its axis of rotation is perpendicular to the direction of the field, determine the induced emf in the coil. N = 50 A = 2 B NB at B = A w - = P.O 60 9.10 Determine the induced emf in the V-shaped loop of Figure 9.25. Take B = 0.6xa, and u = 5a, m/s. Assume that the sliding rod starts at the origin when t=0. R = axxaz V3 dx dt u / 5T. y FIGURE 9.25 For Problem 9.10. = 5T. 53 -y/ do 9.11 A car travels at 120 km/hr. If the earth's magnetic field is 4.3 X 10-5 find the induced voltage in the car bumper of length 1.6 m. Assume that the angle between the earth's magnetic field and the normal to the car is = = ( = is = de = 7,6 L = 9.12 An airplane with a metallic wing of span 36 m flies at 410 m/s in a region where the verti- cal component of the earth's magnetic field is 0.4 Find the emf induced on the airplane wing. = = 9.13 As portrayed in Figure 9.26, a bar magnet is thrust toward the center of a coil of 10 turns and resistance 15 If the magnetic flux through the coil changes from 0.45 Wb to 0.64 Wb in 0.02 S, find the magnitude and direction (as viewed from the side near the magnet) of the induced current. ON A no Mia 10. - -9.14 The cross section of a homopolar generator disk is shown in Figure 9.27. The disk has inner radius P1 = 2 cm and outer radius P2 = 10 cm and rotates in a uniform magnetic field 15 at a speed of 60 rad/s. Calculate the induced voltage. Brush B P2 P1 2 Shaft Copper disk = 4132ml 9.15 generator at MHz is connected to the plates of an air dielectric parallel plate capacitor with a plate area and a separation distance of 0.2 mm. Find the maximum value of displacement current density and displacement current. Se or Area = = D's Vs d Loga: J.D J.2. TT. Eo 50 Js = 10 max S =9.16 A dielectric material with = is placed between the plates of a parallel-plate capacitor. Calculate the frequency at which the conduction and displacement currents are equal. Jo d d & - 9.17 The ratio (conduction current density to displacement current density) is very impor- tant at (high frequencies. Calculate the ratio at, 1 GHz for: (a) distilled water = (b) seawater = = (c) limestone = = S/m) / @ 9.18 In seawater (o = 4 S/m, E = = find the ratio of the conduction to the dis- placement currents at 10 MHz. = 9.19 Assume that dry soil hasjo = = Determine the frequency at which the ratio of the magnitudes of the conduction current density and the displacement current density is unity. J MHZ 600000 HZ9.20 In a dielectric = the conduction current density is given = Determine the displacement current density. E= F = 4900 Para falor de = 708 9.21 In a source-free pregion, H = - A/m. Find the displacement current density. 1 a ax ax r ay - az Ho - - 9.22 An ac voltage source is connected across the plates of a parallel-plate capacitor SO that E = V/m. Calculate the total current crossing a m area placed per- pendicular to the electric field. Assume that the capacitor is air filled. To = It = dxdy= JD 709.23 (a) Write Maxwell's equations for a linear, homogeneous medium in terms of and assuming only the time factor (b) In Cartesian coordinates, write the point form of Maxwell's equations in Table 9.2 as eight scalar equations. a) = - = Es b) D.D = T t = dy t + = dx dy DXE = ) ax az = -JB dx Ex Ey Ez t ax az 2T = J dx dz Hy 9.24 Show that in a source-free region (J = Maxwell's equations can be reduced to two. Identify the two all-embracing equations. Eq Maxwell D 2) - 2) XE = - 3 3) 4) = ITfor l estas mas Eg's de = -JB 2T = at 9.25 Show that fields cos cos tay and sin x sin ta2 do not satisfy all of Maxwell's equations. 1) B = E. az D.B t + = dx9.26 Assuming a source free region, derive the diffusion equation = at = = = ( = PV Eo Eo D.E 2E 9.27 In a certain region, = (2yax + xzay + sin A/m find = Se logo PV (Z) - = dpv +C9.28 In free space, the electric field is given by costa Find the charge density Pv that will produce this field. - Eo Senl Z). 9.29 In free space, H= 10 sin Bx)ay A/m. Find E and B. = - 0 = no D XH = B + = 10 B = Eo 10 + = Ea 1 B. = IT = a EZ a 2x = car + M. B=0,33 11309,739.30 In free space, 50 cos ( - kz)apV/m p Find and H. # = = - P 708 1 - 01044 P 10 # = ap ao az ap IT = J JP JP Ep a = 50 - P =59 - P - B= 50 of = - P - - P dz = ap you KZ) - = K=a,339.31 The electric field intensity of a spherical wave in free space is given by - V/m Find the corresponding magnetic field intensity H. (H 1 do ( = sen dr + dr (esféricas) H.Mo -Br) r 9.32 In a certain region for which o = = J = - Bz)ax (a) Find D and H. (b) Determine B. Go ax to To = 27 70 ? ax i = Z Dx E IT = 60. 709 = GaB - dT = E 109 (or ( 9 B ) = = 2,7.70 (- dz Hy ax = 60 9.33 Use Maxwell's equations to derive the continuity equation. k) - = IT IT a logo9.34 In a source-free region, show that - - ( J + at = - - ME = = E) - - 27 - = DT logo 9.35 Check whether the following fields are genuine EM fields (i.e., they satisfy Maxwell's equa- tions). Assume that the fields exist in charge-free regions. (a) A = 40 sin (wt + (b) cos(wt - (c) (3p2 cot + cos as) sin wt (d) sin sin (wt - 1 2Y 2Y Compo P 1'd = C Be to9.36 Given the total electromagnetic energy show from Maxwell's equations that aw at (E X ds - S = + E . = B. - B. = = + 20 IT = + + + 2T 9.37 In air, E = - ay)az V/m. Find H and a. = 2 rx Z - = - + = +9.38 An AM radio signal propagating in free space has E= - Bz)ax - Determine B and N. 9.40 In free space (pv = show that (cos - sin satisfies the wave equation in eq. (9.52). Find the corresponding V. Take C as the speed of light in free space. at = -uJ 9.41 Retrieve Faraday's law in differential form from - - at dada d que D B T9.42 In free space, the retarded potentials are given by V A = - Wb/m 1 where D.A = + JAZ = av (a) Prove that . A = at (b) Determine E. a) / = X D.A = = IT - = ax - ay 2T - E 9.43 Let = - Wb/m in free space. (a) Find Vand E. (b) Express B in terms of and a) 2 = st w ax = = = 2Z at B =9.45 Determine the phasor forms of the following instantaneous vector fields: (a) - (b) - 2x)az (c) - - Forma do TO As ax -J2X c) = 6 = -T 6 it = = - ay9.46 Find the instantaneous form for each of the following phasors: (a) = (b) = + (c) = J.II Just - 20.C 2 ] 20 ga = D Just ax az JII 9.47 In a source-free vacuum region, If p 1 cos(wt - 3z) A/m (a) Express H in phasor form. (b) Find the associated (c) Determine W. - J3Z == do ap 1 P. =1 P a PHO B a PHO -3 - P Eo PEo -3. - = = a ap WP20 c) n as as pab az 1 P 22 P n Hp a Q Hp a = = 3 -3 Z - at = - I J = u9.48 In a certain homogeneous medium, E = and - + = ax A/m If w = X 109 rad/m, find B and : ax ay oz 2x = = ay J + BZ) C - B ax JW Hs = us JW ax ay oz ax ay 2x = ax 00 ax B = . = Ho = E. B - = J = BQuestão 1 (Valor 4,0 pontos): Considerando a figura 1, suponha que a área de uma única espira de um par seja 100 e que a densidade de fluxo magnético seja constante sobre a área das espiras, mas que se modifique com o tempo como = 4,0 ex = 0,30 Np/s. Determine VR em 1, 10 e 100 S. B y - Figura 1 Questão 2 (Valor 4,0 pontos): o raio de uma espira de metal condutor perfeito no espaço livre, situada no plano xy, aumenta à taxa de 1 m/s. Um pequeno resistor de 2,0 é colocado em uma descontinuidade da espira. Assuma que exista um campo magnético B = T. Determine a corrente induzida na espira, mostre com um desenho a direção do fluxo e o sentido da corrente induzida. Questão 3 (Valor 2,0 pontos): Obtenha a forma fasorial diferencial das equações de Maxwell. INFORMAÇÕES IMPORTANTES A avaliação é sem consulta. Só utilize lápis, borracha, caneta e calculadora cientifica simples. Circule a resposta ao final de cada questão. Boa Prof. Dr. Helder Alves Pereira Ondas e LinhasQuestão 1 (Valor 4,0 pontos): Considerando a figura 1, suponha que a área de uma única espira de um par seja 100 e que a densidade de fluxo magnético seja constante sobre a área das espiras, mas que se modifique com o tempo como B = onde 4,0 = 0,30 Np/s. Determine 1, 10 e 100 s. Figura 1 = / az / C - az - 4= IT = - = - IT = - V - 03.20 V = -Questão 2 (Valor 4,0 pontos): o raio de uma espira de metal condutor perfeito no espaço livre, situada no plano xy, aumenta à taxa - m/s. Um pequeno resistor de 2,0 é colocado em uma descontinuidade da espira. Assuma que exista um campo magnético B = T. Determine a corrente induzida na espira, mostre com um desenho a direção do fluxo e o sentido da corrente induzida. aip TTP - 1. apxaz = 1 = = TTP TT V= RI R 2 Questão 3 (Valor 2,0 pontos): Obtenha a forma fasorial diferencial das equações de Maxwell. maxwell = e 2T = * ] e / = =0 Bs# = - t ] JW = + Dr