3.) In region 1, p0,, and D(2+3+2)Clm2 In region 2, z<o, . In region 2250, a,...
#2 1. Find expansions of in powers of z +1, -1, and a respectively, in O< 12 +11 < 2,0 < 12 - 11 < 2, 12/> 1. 2. Find the Laurent expansion for sin(1/2) in powers of z. Where is it valid? 3. Prove that if f(x) is analytic at zo where it has a zero of order m, then 1/f(x) has a pole of order m at zo.
6. (20 pts.) The plane y-0 separates region 1 (y 0), which is a dielectric materia with c, -3.5, from region 2 (y < 0), which is free space. If the electric flux density in region 1 is given by D,-15a, +22ay -20a, [nC/m'], find D..
2. Shade the region of the complex plane defined by <z +4 + 3i : 3 < 3 < 5,2 EC}. Include the appropriate axis labels and any significant points.
3. About Flux: Suppose F-Ji + 2j-k and s is a region on the plane x + y-z-3. Distinguish the scenario where we can bypass the flux integral by simply computing F. A and the scenario where we have to integrate using the shadow method. Compute the flux. (a) S is a circle of radius 3 on the plane. O F.A Which method is appropriate? Compute the flux: O shadow method. 2 9 on the xy plane. (b) S is...
Using the Divergence Theorem, find the outward flux of F across the boundary of the region D F-2xy2i+ 2x2yj+ 2xyk; D: the region cut from the solid cylinder x2 y2s 4 by the planes z- 0 and z 2 A) 1287 B) 32T C) 64m D) 16T F-2xy2i+ 2x2yj+ 2xyk; D: the region cut from the solid cylinder x2 y2s 4 by the planes z- 0 and z 2 A) 1287 B) 32T C) 64m D) 16T
[132 2 2 3 4 17 marks] Question 4 A plane wave is travelling in a vacuum in the +z-direction with wavenumber k and angular frequency . It is linearly polarised in the x-direction, and has electric field given by E(t, z) Eo Cos(kz - wt)f This wave is normally incident on a perfectly electrically conducting, semi-infinite slab in the region z > 0 and the resulting field in vacuum (z < 0) is a superposition of the incident and...
3 Questions Find: a) The electric flux density vector: D=-EVV, where & is a constant = 10 pF/m. b) The electric volume charge density at any point in the region: p, = V D= divergence ofD e) The total charge enclosed by the specified region: Q= fpdv, dv element of volume d) Find Vx D and show that D is irrotational. In the cylindrical region: 0srs2m, 0s0s7/2 0szslm, the potential field V is given by V=50 2 sin volts 3...
Exercise 3 Let f be an analytic function on D(0,1). Suppose that f(z) < 1 for all z € C and f() = 0. Show that G) . (Hint: use the function g(z) = f(2).)
3 Questions Find a) The electric flux density vector: D-EVV, where s is a constant 10 pF/m. b) The electric volume charge density at any point in the region: p, = V. D = divergence of D c) The total charge enclosed by the specified region: Q ffp,dv, dv element of volume d) Find VxD and show that D is irrotational In the cylindrical region: 0srs2m, 0s0s7/2, 0szslm, the potential field Vis given by V=50 sin volts 3 Questions Find...
3. Suppose f(z) is analytic on a region (i.e. open connected) D. Prove that if Im f(2), the imaginary part of f(z), attains its maximum value in D then f(2) must be constant on 3. Suppose f(z) is analytic on a region (i.e. open connected) D. Prove that if Im f(2), the imaginary part of f(z), attains its maximum value in D then f(2) must be constant on