SEE 2523 2 PART A (Answer 3 questions from this part) Q1 (a) An electrostatic field is said to be a conservative field. By using the appropriate formula and Stoke’s theorem, derive the Maxwell’s equation that relates to this concept of conservation of field. (b) (4 marks) A point charge of 9 µC is located at point P (0, 2 m, 1 m), a uniform line charge with density 170 nC/m lies along the x axis, and a uniform sheet of charge with density 26 nC/m2 lies in the z = 2 plane. (i) Calculate the total electric flux leaving the surface of a sphere of 6 m radius centered at the origin. (ii) (6 marks) Given the points A (-4 m, 3 m, 5 m) and B (5 m, -2 m, 9 m), find the potential difference VAB due to all three charges. (10 marks) (c) If π = π₯ − π¦ + 2π§, find the electrostatic energy stored in a cube of side 2m centered at the origin. (5 marks) SEE 2523 3 Q2 (a) Discuss why Laplace/Poisson’s equation is a more practical approach to solve electrostatic problems compared to Gauss’s Law. (b) (4 marks) Determine which of the following potential field distributions satisfies Laplace’s equation. (c) (i) π1 = π₯ 2 + π¦ 2 − 2π§ 2 + 10 (2 marks) (ii) π2 = ππ§ sin ∅ + π 2 (4 marks) Two large parallel conducting plates are separated by a distance d and maintained at potentials 0 for the lower plate and V0 for the upper plate. The region between the plates is filled with a continuous distribution of electrons having a volume charge density ππ£ = −π0 π¦/π. By assuming negligible fringing effect at the edges, determine (d) (i) the potential at any point between the plates; (6 marks) (ii) the surface charge densities on each plate. (5 marks) By analyzing your answer in Q2c(ii), is it possible to compute the capacitance? If so, what is the value. Explain your answer. (4 marks) SEE 2523 4 Q3 (a) Discuss the effects of the existence of dipole moments when a magnetic material was immersed in a magnetic field. (b) (4 marks) A cross section of coaxial cable centered along the z axis is shown in Fig. Q3. The internal non magnetic conductor with radius a carries a uniform current I Ampere in the direction of −π§Μ . The region π < π < π is a magnetic material with permeability of 4π0 . The outer conductor π < π < π carries a uniform current of I Ampere but in the direction of +π§Μ . (i) Obtain the magnetic flux density in all the regions. (8 marks) (ii) Show that the Ampere circuital law of Maxwell’s equation in point form from Q3(b)(i) is obeyed in the region π < π < π. (iii) (4 marks) Determine the magnetized surface current on the surfaces of the magnetic material. (iv) (5 marks) Determine the force needed to split the outer conductor from the inner conductor. (4 marks) y a z c b Fig. Q3 x SEE 2523 5 Q4 (a) By applying suitable equations, develop the boundary conditions for two magnetic materials having permeability of π1 and π2 as shown in Fig. Q4. Assume the existence of surface current π½π Μ (π΄/π) on the boundary surface. (6 marks) Μ 1 π» π1 π½π Μ π2 Μ 2 π» Fig. Q4 (b) A first region (π₯ < 0 and π1 =3π0 ) is separated from the second region (π₯ > 0 and π1 =π0 ) by a plane of π₯ = 0. Given that a magnetic field intensity in the first region is Μ 1 = 10(π¦Μ + π§Μ )/π0 (A/m). π» A uniform surface current of π½π Μ = 8/π0 π¦Μ (A/m) is conducted on the x = 0 surface. Obtain: (i) the magnetization vector in the first region. (2 marks) (ii) the magnetic field intensity and magnetization vector in the second region. (8 marks) (iii) (c) the energy density in both regions. (6 marks) Will there be any changes in the magnetization vector in both regions if the magnetic Μ 1 = (π¦Μ + π§Μ )π»(π₯). Explain your field intensity in the first region is not uniform; i.e π» answer. (3 marks) SEE 2523 6 PART B (Answer 1 question from this part) Q5 (a) According to Faraday’s experiment, an induced voltage called electromotive force or simply emf can be generated in a closed circuit. State and derive the mathematical expressions for all cases. (b) (6 marks) The square loop ABCD shown Fig. Q5 is immersed in a uniform magnetic field B ο½ 0.5 yˆ Wb/m 2 . The side CD of the loop cuts the flux lines at the angular frequency 120ο° rad/s. If the loop lies in the yz plane at time t = 0, determine: (c) (i) The induced emf at t = 1 ms. (9 marks) (ii) The induced current at t = 3 ms. (3 marks) If everything is the same as in Fig. Q5, except that the magnetic field now is changed to time varying, B ο½ 0.2tyˆ Wb/m 2 , determine the emf at t = 1 ms. (7 marks) (Hint: Unit vector transformation: Cylindrical component of yˆ ο½ sin ο¦ rˆ ο« cos ο¦ ο¦ˆ ) z B 0.03m C R= 0.02m y A D x Fig. Q5 SEE 2523 7 Q6 (a) A uniform plane wave is a particular solution of Maxwell’s equations with πΈΜ assuming the same direction, same magnitude and same phase in infinite planes perpendicular to Μ field. By using the formulas the direction of propagation. This is also applied to the π» obtained for the plane waves in lossy media in terms of permittivity, permeability, and conductivity, determine the phase constant, attenuation constant, and phase velocity of the other three mediums. (b) (6 marks) The electric field intensity of a linearly polarized uniform plane wave propagating in the +z direction in seawater is πΈΜ = 50 cos(2π × 107 π‘)π₯Μ (π/π) at z = 0. The constitutive parameters of seawater are ππ = 72, ππ = 1, π = 4 (π/π), determine: i) The type of medium (2 marks) ii) The phase constant, β and attenuation constant, α (3 marks) iii) The phase velocity, v and intrinsic impedance, η (3 marks) iv) The skin depth and wave length (2 marks) v) The distance at which the amplitude of πΈΜ is 2% of its value at z = 0 (3 marks) vi) Μ (π§, π‘) at z = 1m as function of t. Write the expressions for πΈΜ (π§, π‘) and π» (6 marks)