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Electromagnetic Field Theory Course Description (EC341) Associate. Prof. Dr. Hussein Hamed Ghouz Course Contents Chapter (1): Coordinates systems and vector analysis (3-Lectures) • • • Coordinates systems Vector operations Vector analysis Chapter (2): Static Electric Field (7-Lectures) • • • • • • • • • Column's law for electric force Electric field and electric flux in free-space Gauss' Law, and application to electrostatic field Work Done and Electrostatic Potential Electric Dipole Dielectrics and Polarization Boundary condition of static electric field Basic Equations of Static Electric Field Capacitance Chapter (3): Electrostatic Problems (3-Lectures) • • Image Method Boundary Value Problems and Laplace and Poisson Equations Chapter (4): Currents and Conductors (2-Lectures) • • • • Ohm' Law Joule' Law Resistance Boundary condition of stationary currents density Chapter (5): Static Magnetic Field (6-Lectures) • • • • • • • Basic Equations of Static Magnetic Field Ampere's Law Biot-Savart Law Magnetic Vector Potential Boundary condition of static magnetic field Magnetic Force Inductance ٢ Chapter (6): Time-Varying Fields and Maxwell's Equations (5-Lectures) • • • • • Faraday' Law Displacement Current Maxwell' Equations Time-Harmonic Fields Uniform Plane Wave ٣ Course Plane Chapter No. Chapter Title Chapter 1 Orthogonal Coordinate 1st & 2nd W Systems and Vector Analysis Chapter 2 2nd, 3rd & 4th W Electrostatic Field in Vacuum Chapter 2 Electrostatic Field in Dielectric 5th W Media Chapter 3 Methods for the solution of 6th & 8th W Electrostatic Problems Chapter 4 8th & 9th W Chapter 5 9th, 10th, 11th & 13th W No. of Lectures 3-Lecture 5-Lectures 2- Lectures 3- Lectures Steady Electric Currents 2- Lecture The steady Magnetic Field 6- Lectures Chapter 6 Time Varying Field & 13th ,14th & 15th W Maxwell’s equations ٤ 5- Lectures Exercise and Quiz Test Plane Problem Set No. Problem set # 1 1-Weeks Problem set # 2 3-Weeks Week No. 1st Week 2nd, 3rd & 4th Week Problem set # 3 2- Weeks Problem set # 4 3- Weeks th th 5 & 6 Week 1st Quiz 6th Week 8th , 9th & 10th Week 2nd Quiz Problem set # 5 3- Week Quiz Test 11th, 13th & 14thW eek ٥ 11th Week Coordinate Systems and Vector Analysis ∇ V= VVVV ∇2 = ∂V a h 1∂u 1 u1 + ∂V ∂V a + a h 2 ∂u 2 u 2 h 3 ∂u 3 u 3 ∂ 1 ∂V + ∂ h h 1 ∂V + ∂ h h 1 ∂V h 2h 3 h ∂u ∂u2 1 3 h ∂u ∂u3 1 2 h ∂u h 1 h 2 h 3 ∂u1 1 1 2 2 3 3 1 AAAA ∂ ∂ ∂ (h 2 h 3 A 1 ) + (h 1 h 3 A 2 ) + (h 1 h 2 A 3 ) ∂u 2 ∂u 3 h 1h 2 h 3 ∂u1 1 ∇. = ∇X h1au1 ∂ 1 A= h1h 2h 3 ∂u1 h1 A u1 h 2a u2 ∂ ∂u2 h 2 A u2 ∫ ∇ • A dv = ∫ A • ds v s Divergence Theorem: Stokes's Theorem: h 3au3 ∂ ∂u3 h 3 A u3 ∫ (∇ × A ) • ds = ∫ A • dl s c Metric coefficients of Coordinate systems (x, y, z) (ρ, Φ, z) (r, θ, Φ) au1 ax aρ ar au2 ay aΦ aθ au3 az az aΦ h1 1 1 1 h2 1 ρ r h3 1 1 r sin θ ٦ Transformation between Coordinate Systems ax ay az • • • aρ cosΦ sinΦ 0 aΦ -sinΦ cosΦ 0 az 0 0 1 ar sinθ cosΦ sinθ sinΦ cosθ aθ aΦ cosθ cosΦ -sinΦ cosθ sinΦ cosΦ -sinθ 0 Integral Forms Integral Form Result of Integration ln( x + x2 + a2) sinh −1 xa dx x2 + a2 xdx ∫ x2 +a2 dx ∫ 2 2 3/ 2 (x + a ) ∫ x2 + a 2 x a 2 x2 + a2 dx ∫ 2 2 ( x +a ) −1 x2 + a2 1 tan-1 x a a xdx ∫ 2 2 ( x +a ) 1 ln( x2 + a2) 2 x dx ∫ 2 2 3/ 2 (x + a ) ∫ dx x (x 2 + a 2 ) − 1 ln a 2 2 a + x + a x 1 ln tan ax a 2 dx ∫ sin ax ٧ Differential Calculus Function Differentiation sin(x) cos(x) cos(x) -sin(x) tan(x) sec2(x) 1 sin(x)-1 1−x2 -1 cos(x)-1 1−x2 tan(x)-1 1 1 + x2 ex ex ln(x) 1/x loga(x) 1/[x ln(a)] Trigonometric relations sin(x) = 1/2(1- cos(2x)) cos(x) = 1/2(1+ cos(2x)) tan(x) = (1- cos(2x)) cos( x) sin(2x) = 2 sin( x) cos( x) sin(x ± y) = sin( x) cos( y ) ± cos( x) sin( y ) cos(x ± y) = cos( x) cos( y ) m sin( x) sin( y ) e ± jx = cos(x) ± j sin(x) ٨ Static Electric Field 1. Coulomb’s law: q F2 = k 1 q R2 2 a , Where, k=1/(4πε )= 9x109, and ā is the unit vector in 0 R r the direction of the force ( ar = âr =ār= r / r ). If the force is negative, it is attraction force, while the positive sign means repulsion force. 2. Electrostatic Field and Potential (E and V): r E Point E Line E Surface E Volume = P P P P = = = qi 4 π ε o Ri 2 1 4π ε o ∫ r R V ρ r R3 R d l′ V ρs r R ds ′ ∫∫ 4 π ε o s R3 1 ρv r R dV ′ ∫∫∫ 4 π ε o v R3 1 r Ep = −∇VP V V Vp = − and P P P P = = = = qi 4 π ε o Ri 1 4π ε o 1 ∫ ∫∫ 4π ε o s 1 ∫∫∫ 4π ε o v ρ R d l′ ρs R ρv R ds ′ dV ′ r P r • ∂ E ∫ P l Ref The Ref. point usually have a zero potential 3. Gauss’s law: ∫ D.n ds = Q (total charge enclosed) s Where; Q = ∫∫∫ ρ v dv v = ∫∫ ρs ds s = ∫ ρl dl volume charge surface charge line charge ٩ 2 ∫ D.n ds = 4πr Dr for sphere of radius r s = 2πρ l Dρ for cylinder of radius ρ and length l 4. Boundary Condition of electric field (air-conductor interface): E n ρs = E = 0 t and εo 5. Finite Line charge: L/2 EP = 2 π εoρ 2 2 ρ + ( L / 2) ρl aρ ρ 2 + ( L / 2) 2 + L / 2 VP = ln ρ 2 π εo ρl 6. Infinite Line Charge: EP = ρl a 2 π ε oρ ρ VP = and ρ ln o 2 π εo ρ ρl 7. Electric Dipole: V= p cos θ 1 (2 p cos θ a + p sin θ a ) and E = r θ 2 3 4 π εor 4 π εo r 8. Basic Equations of Static Electric Field ∇.D =ρ ∇ × E= 0 9. Polarization vector P: P=D −ε o E = ( ε − 1) ε E r o 10. Boundary Conditions of Electric Field (dielectric-dielectric interface): E1t = E2t and D1n − D2n = ρs (where, ρs is free charge) 11. Capacitance: C= Q V ١٠ Where; Q is the Gauss’s law ( ∫ D. n ds = ∫∫∫ s v ρ v dv = Q ) V is the line integral of E along the path ( Vab = − int ial = a E . dl = ∫ final = b Va - Vb) To find the capacitance, we follow the 4-steps: 1. Assume a charge +Q and -Q on the two conductors. 2. Find E using Gauss’s law or any other method. 3. Find V using line integral along E lines 4. Put Q = C V, and then Find C. 12. Electrostatic Energy: For N-discrete charge: We = Where, n 1 ∑ Q V 2 k =1 k k 1 V = k 4π ε o n Qj ∑ j = 1 Rjk and j ≠ k For continuous charge: We = 2 1 ε ∫∫∫ ( D . E ) d v ′ = ∫∫∫ E 2 v' 2 v' d v′ For any capacitor configuration: 1 1 Q2 W = C V2 = QV = e 2 2 2C Surface bounded charge for any capacitor configuration: ρUpper = First Conductor = P • (−an ) ρ Lower = Second Conductor = P • ( + a ) n ١١ Solution of Electrostatic Problems 1.Image Method: z z +Q R1 Conducting plane (PEC) h y P +Q R2 y -Q x Fig.1 (a) Fig.1 (b) +Q -Q +Q +Q -Q Fig.2 (b) Fig.2 (a) If the conductor is in the form of corner with an angle “α”, then in this case, we have m images where, m = 2n -1 and n = 180 α 2. Boundary Value Problems In Cartesian: 2 ∂2 ∂2 ∂ 2 ∇ V = + + V =0 P 2 2 ∂z 2 p ∂ x ∂ y ١٢ The possible solutions of W ' ' (u) + k 2u W(u) = 0 k2 u k 0 0 W(u) u Linear Solution Ao u + Bo Periodic Solution + k - jk A1 sin (k u) +B1 cos (ku) C1 e+j k u + C2 e-j k u Decayed Solution A2 sinh (ku)+B2 cosh (ku) D1 e+ k u + D2 e- k u In cylindrical: 2 ∇ V= 1 ∂ ∂V 1 ∂ 2V ∂ 2V (ρ )+ = 0 + ρ ∂ρ ∂ρ ρ 2 ∂φ 2 ∂Z 2 In case of Φ-dependence solution is given as: Φ (Φ) = Ao Φ + Bo In case of ρ-dependence solution is given as: dV(ρ) d (ρ )=0 dρ dρ ⇒ V (ρ) = A1 ln (ρ) + A2 In Spherical: ∇2 V = 1 ∂ 1 ∂V ∂ ∂V (r 2 )+ ( sinθ )= 0 ∂r ∂θ r 2 ∂r r 2 sin θ ∂θ (a) In case of r-dependence only: 1 d dV (r 2 )= 0 2 d r d r r ⇒ V(r, θ, Φ) ≡ V(r) Then: d dV (r 2 ) = 0 dr dr ⇒ (r 2 ١٣ dV ) = A1 dr dV A1 = dr r 2 ⇒ dV = A1 dr r2 -1 V(r) = A1 + A2 r The constant A1 and A2 are determined from the given BC’s. (b) In case of θ-dependence only: d dV ( sinθ )= 0 dθ r 2 sin θ dθ 1 dV A 1 = dθ sin θ Since the integration ∫ ⇒ ⇒ sin θ dV = ∫ dV = A1 dθ A1 sin θ dθ dx 1 ax = ln tan , therefore, the electric potential is sin ax a 2 given as: θ V(θ) = A1 ln tan + A2 2 Where, the constant A1 and A2 are determined from the given BC’s. Currents and Conductors 1. Ohm’s law: In any conducting medium, if an electric field is applied, a current density J is given by: J= σE Where, σ Denotes conductivity of the conductor J Denotes current density A/m2 ١٤ σ = J A/m 2 = = E V/m moh (siemens) A = = S/m V m m The total current passing through an arbitrary surface S of a conducting body is given by I = ∫∫ J . ds s 2. Resistance: R= V I 2 ∫ E . dl 1 R= σ ∫∫ E . ds s → (a) The voltage V is applied along the coordinate u1 while the current is passing across the surface of the plane u2-u3. L R= ∫ o σ du 1 h 2h3 du 2 du 3 h1 ∫∫ (b) Solving Laplace Equation 1. Find the electric potential V 2. Find the electric field as E = − ∇ V 3. Find the current density J = σ E 4. Find the total current I = ∫∫ J . ds s 5. Find the resistance R as R = V I 3. Duality Law: There is a dual relation between the resistance and the capacitance as: ١٥ resistance x capacitance 64748 R x C = ε σ { ratio of the constitutive parameters of the conducting medium Duality between J and D Conductor Dielectric J = σE D = εE ∇. J = o ∇. D = o Jn1 = Jn2 Dn1 = Dn2 J t1 J t 2 = σ1 σ 2 D t1 D = t2 ε1 ε2 σ ε I Q 1 = G R C Static Magnetic Field Basic Equations of Static Magnetic Filed: • ∇ . B = 0 , ∇ x H = J , and B = µ H Ampere's Law • ∇ x B = µo J • ∫ B. dl = µ o I c Differential form Integral form Biot-Savart Law for a line carrying constant current: • H=∫ L I dl x a R 4 π R2 For finite Line carrying constant current: H= I [ sin (the angle in current direction)− sin (the angle in oppsite current direction) ]aˆ n 4πd ١٦ Magnetic Vector Potential for a line carrying constant current: • A = ∫ c µ o I dl 4πR Loop of constant current I and radius r=b: • H center = I ( ±a z ) 4b Boundary Conditions: H1L = H2L B1n = B2n → → H1t = H2t µ1 H1n = µ2 H2n In case of current existing at the boundary surface: H1t - H2t = Js Magnetic Force Fm: F12 = ∫ I 2 d l 2 x B1 c and F = ∫ I dl x B 21 1 1 2 c Magnetostatic Energy: Wm = 2 1 µ ∫∫∫ ( B . H) d v ′ = ∫∫∫ H 2 v' 2 v' d v′ Inductance L: To find the inductance of any geometry, we should do the following steps: 1. Choose the suitable coordinate system, and assume source current I. 2. Find the magnetic field density B (or H) - Ampere’s Law (if there is a symmetry) - Biot-Savart Law - Magnetic vector potential B = ∇ x A 3. Find the magnetic flux ψ as ψ = ∫∫ s 4. Find the flux linkage Λ = ψ N 5. Find L as L = Λ / I ١٧ B . ds Maxwell’s Equations for Time-Varying Fields Differential form ∇xE=− ∂B ∂t ∇ xH =J+ ∂D ∂t Integral form E. dl = − ∫ ∫ ∫∫ ∂B . ds ∂t ∂D . ds ∂t H. dl = I + ∫∫ ∫ ∇.D = ρ D. ds = ∫ ∇.B = 0 ∫∫∫ ρ dv B. ds = o D=εE B=µH J =σE Maxwell’s Equations for complex Time-Harmonic Fields • ∇ x E=− j ω µ H • ∇ x H = − jω ε E + σ E • ∇ . E = ρ/ε • ∇. H =0 • • D=ε E B= µ H • ∫ E • d l = − jωµ ∫∫ B • d s s ∫ H • d l = (− jωµ + σ ) ∫∫ E • d s s ∫∫ E • d s = Q / ε s ∫∫ B • d s = 0 s • • • • • → Differential form → D=ε E B= µ H ١٨ Integral form Problem Set #1 P.1-1 Three corners of a triangle are at P1(0, 1, -2), P2 (4, 1, -3), and P3 (6, 2, 5). a) Determine whether the triangle P1P2P3 is a right triangle. b) Find the area of the triangle. P.1-2 Given the vector A =3ax + 4ay – 6az in cartesian. Express this vector in the following coordinate systems: a) Cylindrical coordinate system. b) Spherical coordinate system. P.1-3 A vector field F is expressed in spherical coordinates as: F = (25/r2) ar. a) Find | F | and Fx at the point P (-3, 4, -5). b) Find the angle that "F" makes with the vector B =2ax - 2ay +az at the point P(-3, 4, -5). P.1-4 Given the vector field function F = x2y ax + xy2 ay , evaluate the scalar line integral ∫ F.dl from the point P1(2, 1, -1) to the point P2 c (8, 2, -1) a) Along the parabola x=2y2 b) Along the straight line joining the two points. π π P.1-5 Given a scalar function V= sin( x) sin( y) e − z . Find the magnitude 2 3 and direction of the maximum rate of increase of V at the point P (1, 2, 3). P.1-6 For the vector function A =ρ2aρ + 2z az, verify the divergence theorem for circular cylinder region enclosed by ρ=5, z=0, and z=4. ١٩ P.1-7 A vector field A =ar (cos2Φ)/r3 exists in the region between two spherical shells defined by r=1 and r=2. Evaluate: b) ∫ ∇ . A dv a) ∫s A . ds v P.1-8 Given a vector field A =3x2y3 ax - x3y2 ay. Verify stokes theorem for the contour shown in figure. y 2 S 1 x 1 HW: P1.1, P1.3, P1.5, P1.11, P1.12, P1.15, P1.29, P3.29, P3.30, P4.5 ٢٠ 2 Problem Set #2 P2-1 Two charges q1 = +2 mc, q2 = -5 mc are located at points (2,0,0) and (-2,0,0) respectively. Find the force acted on a third charge q3=+3mc if it is located at (0, 0, 6). Find E at P (3, 3, 8). P2-2 Consider a charged dielectric sphere of radius a = 0.5m and ρv=2mc/m3. Find the electric field at r = 0.2 m, r = 0.5m, and r=2m. Plot the variation of the electric field versus r. P2-3 A coaxial line has an inner conductor of a radius “a” and outer conductor of a radius “b”. The inner conductor is charged by +ρℓ while the outer is charged by -ρℓ. Find electric field as function of ρ, and then find the potential difference between outer and inner conductors. P2-4 A circular ring has a radius a=2m lies in z=0 plane with its center at the origin. If ρℓ =10 nc/m. Find a point charge at the origin which could produce the same electric field at P (0, 0, 5). P2-5 Two uniform infinite line charges ρℓ= 4 nc/m lies at x = 0, y=+4 and parallel to z – axis. Find the field at (4, 0, z). P2-6 Show that the electric field is zero inside conducting sphere charged by Q=5 nc and has radius 5=5 m. Find E at r=10 m. P2-7 Find the work don-e to move Q = 5 µc from the origin to the point P (2, π/4, π/2) in an electric field: E = 5 e-r/4 ar + ٢١ 10 a r sin θ φ V/m. P2-8 Prove that the potential of a single infinite line charge (consider a reference point at ρ=ρo has a zero potential) V = ρl ρ ln ( o ) . 2 π εo ρ P2-9 Three uniform finite line charges + ρl , ρl , and ρl each of length 1 2 3 “L” forming an equilateral triangle. Assuming that ρl =2 ρl = 2 ρl . 1 2 3 Determine E at the center of the triangle. HW: P2.11, P2.18, P.2.22, P2.26, P3.2, P3.9, P3.15, P3.20, P4.2, P4.6, P4.8, P4.16, P4.33, P4.35 ٢٢ Problem Set#3 P.3.1 Find the capacitance of the following structures: (a) Two parallel plates: A A1 A2 d d1 ε1 ε2 ε1 d2 ε2 (i) (ii) (b)Coaxial lines: b ε1 ε1 a ε2 ε2 b a c (i) (ii) (c) Concentric spheres: ε1 R2 R1 R1 = a R2 = b ε2 ٢٣ P.3.2. Find the capacitance between two identical and long cylindrical conductors of radius "a" as shown in the following figure. These conductors are separated by air, and the distance between their centers is "d". a a εo ρℓ -ρ ρℓ d P.3.3. Given that E1 = 2 a x − 3 a y + 5 az v/m as shown in the following figure. Find D2 and the angles θ1 and θ2. z E1 θ1 ε1=2εo x-y ε2=5εo E2 θ2 HW: P6.5, P6.7, P6.10, P6.12, P6.13, P6.16, P6.19, 6.31 ٢٤ Problem Set #4 P.4.1 Find the potential V inside the following drawn wells: y y V=0 ∞ ↑ b V=0 V=0 -∞ ← x V=Vo x d V=0 P.4.2 Find V between the two planes φ = 0 and φ = V = 0 at φ = 0 and V = Vo at φ = π if the potential 4 π . 4 P.4.3 Find V between two coaxial cones if V = o θ= V=Vo at θ = π V = Vo at 4 π 10 P.4.4 A straight conducting wire of radius “a” is parallel to and at height “h” from the surface of earth as shown in the following figure. Assuming the earth is perfectly electric conducting; determine the capacitance and the force per unit length between the wire and the earth. a h P.4.5 Find the resistance of the different configurations given in P.3.1 of problem set#3, assume lossy media. [Hint: use duality equation R × C = ε ]. σ ٢٥ P.4.6 Find the resistance of only one of the different configurations given in P.3.1 of problem set#3, assume lossy media. [Hint: use Laplace equation]. HW: P7.6, P7.10, P7.15, ٢٦ Problem Set #5 P.5.1 Find the magnetic field at point P due to the stationary currents going through the following wire configurations: b P a a I I b P P.5.2 Find the flux inside a rectangle of dimensions a x b near a straight wire with current I as shown in Fig.1. P.5.3 Find the magnetic flux density B of a wire carrying constant current I1 at the point P as shown in Fig.2. Assume I1=0.5 mA, d=0.3m, and the loop radius b=0.05m. P.5.4 Find the inductance of the following magnetic coil shown in Fig.3. P.5.5 Consider a transmission line of two long parallel conducting wires of radius “a” as shown in Fig.4. Assume that the two wires are located in x-z plane and they carry equal currents in opposite direction. Find the internal inductance, mutual (external) inductance and the magnetic force. Fig. 2 Fig. 1 y I1 d d I b d b a P x I1 ٢٧ y Fig. 4 Fig. 3 I I S I a N x d z P.5.6 Given E (t, z) = Eo sin (ω t – βz) a x , find D and H in the free space. Sketch E and H at t = 0. P.5.7 An electric field component in y-direction propagates in the freespace along the z-direction. The electric field intensity has an instantaneous form given by: j[ω t − (π / 3) z] E (t, z) = Re (Eo e o ay) Find D (t, z) and B (t, z). HW: P8.5, P8.9, P8.12, P8.25, P8.27, P8.36, P8.41, P10.15, P10.18, P10.26 ٢٨ COLLEGE OF ENGINEERING & TECHNOLOGY Department: Electronics and Communications Engineering Instructor: Dr. Hussein Hamed Ghouz Course Title: Electromagnetics Course No.: EC341 Marks: 30 Date: Tue., May, 3, 2011 Time: 60 Min Answer the following questions: (Version_A) Question No. 1 (a) Given Two vectors A=4ax+3ay+5az and B=5aρ+4az with an angle Φ=36.87o. Find the cross and dot product in cylindrical coordinate. Find the angle between A and B (b) Given a vector of flux density D=xy2ax+4xyay C/m2. Verify the divergence theorem within a parallelepiped formed by planes 0≤ x ≤2, 0≤ y ≤3, and 0≤ z ≤4. (c) Find the curl of the vector D Question No. 2 Three point charges Q1=10 nc, Q2=-5 nc, and Q3=-10 nc are located in space at the points P1 (xo,0,0), P2 (0,yo,0), and P3 (0,0,zo) respectively. Find the following: (a) The electric force acting on Q3 assume xo=0, and yo=zo=1.0 (b) The electric field and potential at the point P (xo, 2yo, 3zo) (c) The electric energy We Question No. 3 Given a Charged Disk of radius b = 50 cm, and charge density ρs = 50 nc/m2 is located in x-y plane as shown in the following figure. Find the following: (a) The electric field at the point P (b) The electric field at the point P if b → ∞ (c) Verify the results of part (b) using Gauss' Law Formula Sheet ∫∫ (∇ × D) • ds = ∫ D • dl S c ∇×D = ( ∂D z ∂y − ∫∫ D • ds = ∫∫∫ ∇ • D dV S V ∇•D = ∂D x ∂D y ∂D z + + ∂x ∂y ∂z ∂D y ∂D y ∂D ∂D ∂D x x )a )a − ( z − )a + ( − x y ∂z ∂x ∂z ∂x ∂y z x dx = + a2 )3/ 2 a2 x2 + a2 xdx −1 ∫ (x 2 + a 2 ) 3 / 2 = x 2 + a 2 ∫ (x z 2 P (0, 0, z) ρ=b GoodLuck ٢٩ x y COLLEGE OF ENGINEERING & TECHNOLOGY Department : Electronics and Communications Engineering Instructor : Dr. Hussein Hamed Ghouz Course Title : Electromagnetics Course No.: EC341 Marks: 20 Date : Sun., May, 29, 2011 Time: 60 Min Answer the following questions: (Version_A) (a) Given a cylindrical capacitor (b=2.7a, Vo=5 & ԑ2=4ԑ1) shown in Fig. 1. Using Gauss' law, find the following: 1. The electric field vector E and the polarization vector P in each region 2. The electric energy We , and the bounded charge densities on inner and outer conductors 3. The capacitance in each region (b) Given two, isolated and infinite conducting planes form a sector of an angle α=60o as shown in Fig.2. One of the conducting planes is kept at constant potential Vo=10 volt while the other is grounded. Solve the Laplace equation to find the the following: 1. The potential distribution V between the conducting planes 2. The electric field distribution E between the conducting planes 3. The capacitance (c) Given two ideal dielectric regions as shown in Fig.3. The electric field in the first region is E1= 4aρ+2az with an angle Φ=30o. Using the boundary condition of the electric field, find D1, D2, α1 & α2 z ε2 y E1 V=Vo α1 V=Vo 2a ε1=2ε =2 o ε2=4ε =4 o α=60 0 x Fig.2 b Fig.3 V=0 Fig.1 AAAA ∇ V = = ∂V h ∂u 1 1 a u 1 + ∂V h ∂u 2 2 a u2 + ∂V h ∂u 3 3 a u 3 ∂ ∂ ∂ ( h 2 h 3A1 ) + ( h1h 3A 2 ) + (h h A ) h1h 2 h 3 ∂u1 ∂u 2 ∂u 3 1 2 3 ε We = ∫∫∫ ( E . E ) d v ′ 2 v' 2 ∇ V = E2 V=0 ε1 1 GoodLuck ٣٠ α2 ρ COLLEGE OF ENGINEERING & TECHNOLOGY Department: Electronics and Communications Engineering Lecturer: Associate Prof. Dr. Abd-El-Hamid Hamid Gafer Associate Prof. Dr. Hussein Hamed Ghouz Course: Electromagnetics Electromagnetics-I Course Code: EC341 Date : Thr. 30, June, 2011 Time : 120 Min Total Marks : 40 ________________________________________________________________ Answer All Question Question No. 1 (8 Mark) (a) A lossy material (ԑ=4.6ԑo, σd=3x10-2 & µ=µo) is used to fill the space between the inner and outer conductors of a coaxial cable as shown in the figure. The inner conductor is kept with a positive voltage Vo=5 volt and the outer conductor is grounded. Assume,, the thickness of the outer conductor can be neglected, find the following: following 1. The capacitance C and the he inductance L per unit length 2. The conductance G=1/R per unit length Vo b 3. Draw the equivalent circuit (b) Find the coaxial cable radii ratio to achieve characteristic impedance a ԑ, µ, & σd L Zo= =50 Ω C Question No.2 (10 Mark): (a) Consider a finite line of a length ℓ=0.5m carrying a stationary current I1=100 mA as shown in the figure. The line is located symmetrically on y-axis at b=0.5m and parallel to-z-axis. to Find the magnetic field intensity at the point P(x, 0, z) (b) Find the magnetic flux density B of a line conductor configuration carrying constant current I2 at the point P as shown in the figure. Assume I2=50 mA and d=0.3m.. y x d P(x, 0, z) I2 (a) 2d P y (0, b, 0) x (b) I1 d z I2 2d Question No.3 (8 Mark) (a) Write down the set of Maxwell’s equations in differential form for Complex Exponential Time-Harmonic fields Harmonic electric field having an x-component propagates in a general lossless (b) A Time-Harmonic meduim (ԑ=4.6ԑo& µ=µoµr) along the z-direction.. The electric field intensity has an instantaneous form given by: π π 4 3 E (t, z) = 10+03 sin ( 2 π × 10 + 07 t − z + ) ax Find D (t, z), B (t, z) and µr P.T.O. ٣١ Question No.4 (10 Mark) (a) Given two isolated and infinite conducting planes form a sector of an angle α=60o as shown in figure. One of the conducting planes is kept at constant potential Vo while the other is grounded. Solve the Laplace equation to find the potential and the electric field distributions between the conducting planes (Hint: V is function of the angle φ) y ∇2 V = ∇ V= 1 ∂ ∂V 1 ∂2V ∂2V (ρ ) + + ρ ∂ρ ∂ρ ρ 2 ∂φ 2 ∂Z 2 ∂V a ∂ρ ρ + ∂V a ρ∂φ φ + ∂V ∂z a V=0 α=600 z x V= Vo (b) Given a corner shape of infinite copper conducting planes having a zero potential and an angle θ=90o as shown in figure. Assume a point charge Q= 20 nC is located at the mid-point between the two conductors (x=d, z=d, d=0.5m). Use the image method to find the following: 1. The total number of images to replace the ground planes 2. The electric potential and electric field at observation point P(x, 0, z) P(x, 0, z) z d y→ ∞ d θ x Question No.5 (4 Mark) Two identical circular conducting plates each of radius rc=2.5mm form a parallel plate capacitor as shown in figure. Find the following: 1. The electric field intensity, electric flux density, and polarization vectors 2. The capacitance 3. The bounded surface charge on upper and lower conducting plates z Vo=5 rc Constants h=20 mm 1/(4πεo) = 9x109 m/F εo=10-09/(36xπ) F/m µo=4πx10-07 H/m ε=2.5εo y V=0 GoodLuck x ٣٢