Electromagnetic

Engineering

Electromagnetics

PAAA

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Answer all the Questions. Show your work clearly. Q1. Indicate True (T) or False (F), or circle the answer closest to the correct answer: j (1) 2 pts The phase angle of the quantity: (a) 0o ;  −8 e 4 3− j 3 (b) - 90o ; is (c) 180o .   (2) If A = cos x e − 2 y ln z aˆ z , then  . A is 2 pts (a) − sin x e − 2 y ln z ; (b) −2 cos x e −2 y ln z ; (c) cos x e − 2 y / z . (3) A distortionless line (DL) and a lossless line (LL) have the following property: 2 pts (a) Both have the same β ; (b) Both have the same Z0 ; (c) Both have same β and Z0 . (4) From the input impedance of a short-circuited and an open-circuited line we may find: 2 pts (a) input resistance Rin ; (b) input resistance Rin and reactance Xm ; (c) the capacitance and inductance on the line. (5) If an electrostatic potential V = r2 cos ϕ , the electric field 2 pts (a) sin  aˆr + 2r cos  aˆ z ; (b) 2 r cos  aˆr − r sin  aˆ ; E is: (c) sin  aˆr + r 2 sin  aˆ . (6) If the magnetic field B = a e - 3y â x + b3 cos x â y + 5 c z â z , then a condition that the coefficients a, b, and c must satisfy is: 2 pts (a) -3 a + b 3 = 0 ; (b) -3 a + 5 c = 0 ; (c) -3ab3c = 0 ; (d) c = 0 . (7) The propagation constant and characteristic impedance of a lossless line with L = 36 H/m, C = . 1 F/m at 10 Mrad/s are given by: 2 pts (a) 6 rad/s ; 60 Ω (b) 60 rad/s ; 6 Ω (c) 6 rad/s ; j 60 Ω (d) j 6 rad/s ; 6 Ω . (8) The phase velocity in a distortionless transmission line with dielectric εr = μr = 3 is: 2 pts (a) (1/3)  108 m/s (b) (1/ 3 )  108 m/s (c) 108 m/s . 2 pts (9) Electric polarization is less frequent in materials than magnetization. (10) The derivative operator and the Laplacian operator are respectively: 2 pts (a) both vectors ; Q2. (b) vector and scalar ; (c) scalar and vector . Suppose the magnetic vector potential A and electric scalar potential V at a point in a certain medium (, ) are given by :   A = A0  − x sin  2    z  aˆ x + y sin   2   z  aˆ y    , and V = V0 ( xy + yz + zx ) , where A0 and V0 are constants. 2 pts 10 pts 8 pts Verify if A satisfies the Coulomb gauge. Find the static magnetic field B and the phasor electric field E at the above point. Show your work carefully. (iii) Hence find B and E at the location (2, -2, -4). Assume A0 = 10 , V0 = 20. (i) (ii) Q3. Suppose the vector magnetic potential for a current distribution in the 10 spherical domain (R, , ) is given by: A = 2 aˆ . R sin  2 pts (i) Show that the above satisfies the Coulomb condition for static vector potential, i.e.,   A = 0. 5 pts (ii) Using the spherical curl operator, evaluate the magnetic flux density B for the 5 pts above current distribution. Show your work clearly. (iii) If the relative permeability of the magnetic medium (r) is 150, find the magnetic field intensity H at the location (10, 5/8, /8). Show your work. 8 pts (iv) Find the amount of magnetic energy in Joules stored in a spherical region given by R in the range (1, 10); θ in the range (450, 900); ϕ in the range (0, 900). Show all your work. Note the spherical volume element: dv = R2 sin θ dR dθ dϕ. Note also  csc d = − ln csc + cot  + C, where C is a constant. Q4. (a) Two vectors are given as: E = − 2 aˆx + 7 aˆ y − 3 aˆz , and F = 8 aˆx − 6 aˆ y − 5 aˆz . 14 pts Find a vector G such that it is (i) perpendicular to both E and (ii) parallel to F . Assume G = a aˆ x + b aˆ y + c aˆ z . Show your work in detail. Calculate a, b, c. [Note that for perpendicular vectors, the dot product vanishes. And for parallel vectors, the dot product is the product of their magnitudes. To solve the two simultaneous equations with three unknowns, assume a = 10.] (b) Carry out a dimensional analysis of the following quantities and express the results in terms of MKS units: 6 pts (i) E  H ; (ii)  / β ; (iv) V . Q5. (a) Find the energy (in Joules) stored in a 50 μF capacitor with a voltage of 25 V across it. How much current must a 0.8 mH inductor draw in order to store the same energy as the capacitor? How much is the magnetic self-flux in the inductor that self-flux = LI ). 8 pts (b) 4 pts Consider the complex numbers (i) A = - 6 + j 8 , B = - 7 - j 12 , Write down the magnitude and phase angle of the number: D = AB/C 4 pts (ii) Write down the magnitude and phase angle of the number: E=e 4 pts jD (iii) Now calculate the following: F = M A cos  D + M B sin  E , where M A and M B are the magnitudes of A and B. Show all your work. C = 25 e - j  / 6 ...
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