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GENERAL SIR JOHN KOTELAWALA DEFENCE UNIVERSITY Faculty of Engineering Department of Electrical, Electronic and Telecommunication Engineering BSc Engineering Degree Semester 4 Examination – December 2013 (Intake 29 - EE/ET) ET 4042 – FIELDS AND ELECTROMAGNETICS Time allowed: 3 hours 17 December, 2013 ADDITIONAL MATERIAL PROVIDED Laplace Transform Table 4 cycle semi-log paper INSTRUCTIONS TO CANDIDATES This paper contains 6 questions on 7 pages Answer any FIVE questions only This is a closed book examination Use separate answer book for SECTION A and SECTION B This examination accounts for 70% of the module assessment. A total maximum mark obtainable is 100. The marks assigned for each question and parts thereof are indicated in square brackets If you have any doubt as to the interpretation of the wordings of a question, make your own decision, but clearly state it on the script Assume reasonable values for any data not given in or provided with the question paper, clearly make such assumptions made in the script All examinations are conducted under the rules and regulations of the KDU Page 1 of 8 This Page is Intentionally Left Blank Page 2 of 8 εo = 8.84 x 10-12 F/m μo = 4π x 10-7 H/m SECTION A Question 1 (a) State Gauss Law in electrostatics. [03] (b) A co-axial cable has a core radius r1 and sheath radius r2. The space between the core and the sheath is filled with a dielectric material having a relative permittivity εr. Applying Gauss Law, show that expression (1.1) gives the capacitance per length l of the cable. [05] (c) The co-axial cable shown in Figure Q1.1 has two concentric layers of dielectric between the core of radius r1 = 1 mm and the sheath of radius r3 = 3 mm. The dielectric layers separate at radius r2 = 2 mm. The relative permittivity of inner dielectric εr1 = 4 and that of the outer dielectric εr2 = 3. Length of the cable is 100 m. The voltage applied between the core and the sheath is 1.5 kV and the sheath is grounded. Determine, (i) The maximum electric field intensity in each layer of dielectric [06] (ii) Capacitance of the cable [04] (iii) Total energy stored in the dielectric medium of the cable. [02] εr2 εr1 r3 r2 r1 Figure Q1.1 Page 3 of 8 Question 2 (a) Figure Q2.1 shows a permanent magnet excited circuit to set up a flux density of 0.6 T in the air gap. The circuit has a uniform cross sectional area throughout, an air gap length of 2 mm and a mean iron length of 180 mm. The relative permeability of iron is 2000. Demagnetizing data for the permanent magnet is given in Table Q2.1. Determine the length of the permanent magnet required [10] Table Q2.1 . Bm (T) Hm (kA/m) 0.8 0 0.7 -10 0.6 -15 0.5 -20 0.4 -24 0.3 -27 0.2 -30 0.0 -35 Air gap Mean flux path Mild Steel core Permanent Magnet Figure Q2.1 (b) Figure Q2.2 shows an electromagnetic actuator of circular cross section with inner and outer diameters d1 and d2. The exciting coil has N turns and the mild steel used in the circuit can be assumed to have infinite permeability. At the position shown the air gap length is x. Derive an expression for the force F exerted on the work piece when a current I is sent through the coil. [10] x Mild steel core Coil of turns N Movable mild steel plunger d2 d1 Work piece Force F Figure Q2.2 Page 4 of 8 Question 3 (a) Figure Q3.1 shows two lossy dielectric media of permittivities εr1 & εr2 and resistivities ρ1 & ρ2, separated by a plane boundary. Electric field intensity E1 in medium-1 reaches the boundary at an angle α1 to the normal. Derive expressions for, (i) Outgoing electric field intensity E2 in medium-2 [04] (ii) Direction angle α2 of the field intensity in medium-2 [04] (iii) Surface density ρs of free charges appearing on the boundary [04] E1 α1 ρs α2 εr1 ρ1 εr2 ρ2 E2 Figure Q3.1 (b) A point charge Q is in a medium of relative permittivity εr, at a distance d away from an infinite plane conducting boundary. Derive an expression for the force acting on the charge due to the induced charges on the boundary. [04] (c) What is image method of field calculation in electrostatics? [02] Suggest three practical systems that can use image methods for the calculation of capacitance, conveniently. [02] Page 5 of 8 Question 4 (a) (b) Derive an expression for the self inductance per unit length of a twin overhead line of conductor radius a, and conductor separation D. [07] Figure Q4.1 shows two concentric coils of radii r1 and r2 having no. of turns N1 and N2 in a medium of relative permeability μr with an axial separation d. Derive an expression for the mutual inductance between the two coils. You may assume that coil-2 radius r2 is very small. [07] N1 turns Radius r1 μr N2 turns Radius r2 X X d Figure Q4.1 (c) Derive an expression for the magnetic field intensity at the centre of a rectangular coil of side lengths 2a and 2b, having N turns and carrying current I. Page 6 of 8 [06] SECTION B Question 5 (a) State Maxwell’s equations in electromagnetic field theory. [08] (b) Using Maxwell’s fourth equation, derive an expression for the Amperes Law in stationary conditions of electric field. Hence, show that the total current component is transverse or solenoidal. [04] (c) Equations (5.1), (5.2), (5.3) and (5.4) are given: Poisson equation in free space: Continuity Equation: Lorentz condition for electromagnetic field theory in free space: Vector potential in free space in cylindrical coordinates: Symbols , , and denote the scalar potential, charge distribution, current density and vector potential respectively. The initial state of the scalar potential is zero. Q6 (a) (b) (i) Obtain an expression for the scalar potential as a function of space and time [03] (ii) Obtain an expression for the charge distribution of the system. [03] (iii) Show that it is impossible to obtain a total non solenoidal current with non zero value of with the conditions in (c) (ii) above. [02] Use the Maxwell’s equations in electromagnetic field theory to explain the basic laws of electricity and magnetism. [08] The vector representation of the electric field in cylindrical coordinates of a coaxial transmission line in free space is given by equation (6.1), where , , and are real constants. (i) Obtain an expression for the magnetic field of the transmission line as a function of space and time. Show that the electric and magnetic fields are perpendicular to each other. [04] (ii) Obtain an expression for the current density of the system. Check whether the current component is solenoidal or irrotational. [06] Page 7 of 8 (iii) Obtain an expression for the charge distribution of the system. End of question paper Page 8 of 8 [02]