1. a) State Biot-Savarts law. b) 3. The y - and z-axes, respectively, carry filamentary currents 10 A along ay and 20 A along -az. Find Hat (-3,4,5). 2. A conducting filament carries current I from point A(0, 0, a) to point B(0, 0, b). Show that at point P(x, y,0), a) State Ampere's circuit law. b) A hollow conducting cylinder has inner radius a and outer radius b and carries a current I along the positive z - direction. Find H everywhere. 4. An infinitely long filamentary wire carries a current of 2 A along the z-axis in the +z - direction. Calculate the following: a) B at (-3,4,7) b) The flux through the square loop described by 2 ≤ p ≤ 6,0 ≤z ≤ 4, = 90° 5. Determine the magnetic flux through a rectangular loop (a × b) due to an infinitely long conductor carrying current I as shown in Figure 7.1. The loop and the straight conductors are separated by distance d./nFigure 7.1.- For Problem 5 6. A brass ring with triangular cross section encircles a very long straight wire concentrically as in Figure 7.2. If the wire carries a current I, show that the total number of magnetic flux lines in the ring is 4 Molh 2лb [b-anª+b] Calculate if a = 30 cm, b = 10 cm, h = 5 cm, and I = 10 A. Brass ring Figure 2.- Cross section of a brass ring enclosing a long straight wire; for Problem 6/n7. Consider the following arbitrary fields. Find out which of them can possibly represent an electrostatic or magnetostatic field in free space. a) A = y cos ax ax + (y + e¯x)az 20 b) B=ap c) C = r² sin a 8. Reconsider Problem 7 for the following fields. EECE 3101 Homework 5 a) D = y²zax + 2(x + 1)yzay − (x + 1)z²az b) E= (z+1) Р cos pap + az sin o Р c) F=(2 cos 0 a, + sin 0 a.) Fall 2023/n9. For a current distribution in free space, A = (2x²y + yz)ax + (xy² − xz³)ay - (6xyz – 2x²y²)a₂ Wb/m a) Calculate B b) Find the magnetic flux through a loop described by x = 1, 0<y<2, 0<z< 2 c) Show that V. A = 0 and V. B = 0. 10. The magnetic vector potential of a current distribution in free space is given by A = 15e º sin pa, Wb/m Find Hat (3,7/4,-10). Calculate the flux through p = 5,0 ≤ ≤ π/2,0 ≤ z ≤ 10.

3.4 If r = xa, + ya, + za, the position vector of point (x, y, z) and r = r, which of the following is incorrect? (a) Vr = r/r (b) V.r = 1 (c) √²(r.r) = 6 (d) Vxr=0

3. The optical power after propagating through a fiber that is 450 m long is reduced to 30% of itsoriginal value. Calculate the fiber loss a in dB/km.the tfiber at

Based on wave attenuation and reflection measurements conducted at 1 MHz, it was determined that the intrinsic impedance of a certain medium is ηc = 28.1e45 and the skin depth is 5 m. Determine the conductivity of the material, the wavelength in the medium and the phase velocity.

2. The time-domain expression for the magnetic field of a uniform plane wave traveling in a nonmagnetic medium is given as H(x, t) = 20.2 cos(6π × 108 t – 10.2 x) Find (A/m) (a) the direction of wave propagation; (b) the relative permittivity e, and the intrinsic impedance n; (c) the time-domain expression of the associated electric field E; (d) the time-average power density Sav; and (e) the net time average power of the EM wave crossing a rectangular area of 30 × 20 cm oriented along yz plane, that is, the flat area has a normal vector î.

Problem 2 The electric field of a plane wave propagating in a medium is given by E = [ŷ3 sin( × 107 -0.2x) + 24 cos( × 10² -0.2лx)] (V/m) 1. Find the propagation velocity of the field. 2. The medium has permeability o. Find its relative permittivity €. 3. Is the wave a plane wave? 4. Use both time-domain and phasor-domain Maxwell's equations to find the magnetic field H. 5. Determine the average power density carried by the wave.

6. A brass ring with triangular cross section encircles a very long straight wire concentrically as in Figure 7.2. If the wire carries a current I, show that the total number of magnetic flux lines in the ring is 2mb [b-ana+b 4 = Calculate if a = 30 cm, b = 10 cm, h = 5 cm, and I = 10 A. Brass ring Figure 2.- Cross section of a brass ring enclosing a long straight wire; for Problem 6

2.27 Consider the imaginary rectangular box shown in Fig. P2.27. A wave traveling in the medium has electric and magnetic fields E = 100e -20y cos (27 x 10°t - 40y) (V/m), H = -20.64e-20y cos(2 x 10°t-40y-36.85°) (A/m). The box has dimensions a = 1 cm, b = 2 cm, and c=0.5 cm. Determine: (a) the net time-average power entering the box, (b) the time-average power exiting the box, and (c) the time-average power absorbed by the box. Figure P2.27: Imaginary rectangular box of Problem 2.27.

Q1 A hollow spherical conductor of internal radius R₂ and external radius R3 surrounds a conductive sphere of radius R₁, which is charged with a charge Q,as shown in Figure Q1. Derive the expression for the magnitude of the electric field E(r), for between 0 and ∞o. Note that r = 0 is the origin of the spherical reference system, as shown in Figure Q1.[16] Hence or otherwise, derive the expression for the scalar potential V(r),for r between 0 and ∞. ing R₁Give a qualitative graphical representation for the functions E(r)(magnitude of the electric field) and V(r) (scalar potential), consider-160 cm, and Q = 80 × 10-¹2 C.Use the appropriate units on the graphs. Write the values of the electric field in the dielectric side of the interface for each of the metallic 60 cm, R₂ =100 cm, R3:=surfaces. Write the values for the potential at r = 0, r = R₁, r = R₂, [12]and r = R3.= Briefly describe what happens to the scalar potential and electric field in the region with R₁ < r < R₂ at the static equilibrium if a charged sphere of radius Rs = 50 cm and charge Qs = 2 × 10-6 C is positioned close to the hollow conductor, with its centre at R₁ == 4 m. Include a [10]brief explanation for your answer.

2.40 A horizontal sand layer of thickness d overlays a semi-infinite medium of wet soil. If Esand = (3-j0.1) and Esoil = (20-j5), compute and plot the magnitude of the reflection coefficient, as a function of d from 0 to 1 m, at normal incidence at 2 GHz.