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Information which may be useful: B t HJ f D t D f E B 0 ( F )d F da ( F) da F dl v s v Spherical l p field o o S o E B S T (U Umech ) t field (p p ) t field mech (r): Coordinates F 1 2 1 1 F 2 r Fr Sin( ) F r Sin( ) r Sin( ) r r Cylindrical (rz): Coordinates 1 Fz F Fr Fz 1 (rF ) 1 Fr ˆ ˆr zˆ F + r z z r r r r D o E P H 1 o B M linear media : D E, B H o 8.854 10-12 C/ Nm 2 o 4 10-7 N / A 2 1 1 2 2 2 2 k˜ 1 1 i 1 1 2 Page 1 (Each question is worth 20 marks) Question 1. a) An electric field is applied to a dielectric sphere of radius a, which induces a polarisation of P(r) = r rˆ (for 0≤r≤a), where rˆ is a unit vector in the radial direction and r is the radial coordinate. Using spherical coordinates calculate the bound charges, b [C/m2] and b [C/m3]. [8 marks] b) Given that the relative permittivity of the material is, r = 2.5, calculate the electric field E(r) and electric flux density D(r) inside the material. [4 marks] c) An infinitely long circular cylinder of diameter, a, carries a magnetization parallel to its axis of M = r2 zˆ (for 0≤r≤a), where zˆ is the unit vector parallel to the cylinder axis and r is the radial coordinate from the centre of the cylinder. Calculate the bound current densities Jb [A/m2] and Kb [A/m]. [8 marks] Page 2 Question 2. Consider a rectangular waveguide constructed from an infinitely conductive material with dimensions a and b as shown below. a) Draw a ray diagram in the y-z plane, which illustrates how the wave fronts propagate in the waveguide. Show graphically (or with another method) whether the phase and group velocities are greater than or less than c. [5 marks] b) Given that a = 2.80 cm and b = 2.20 cm and you only want to excite one TE mode, what range of frequencies could you use? [5 marks] Page 3 ˜ (r,t) E˜ eik I rt , c) An electromagnetic monochromatic plane wave given by E I 0I incident on a flat surface gives rise to reflected and transmitted waves of the form, ˜ (r,t) E ˜ eik R rt and E ˜ (r,t) E˜ ei kT rt respectively. Assuming E R 0R T 0T that the polarization of the wave is s-polarized (perpendicular to the plane of ˜ E˜ yˆ , E ˜ E˜ yˆ and E ˜ E˜ yˆ . Show that incidence) such that E 0I 0I 0R 0R 0T 0T 1 E˜ 0R r E˜ 0I and E˜ 0T t E˜ 0I , where r 1 cos( T ) and cos( I ) and t 2 , where 1 1 2 . [10 Marks] 1 2 Page 4 Question 3. a) What is dispersion? [2 marks] b) Show that the phase and group velocity of light in vacuum equals the speed of light c. How does this relation differ in a dispersive material? [3 marks] c) Due to vibrations of electrons in the lattice of a dielectric, the complex permittivity may be written as: fj Nq 2 ˜ r 1 2 m o j j 2 i j Explain in detail what you know about the physics described by this equation with regards to a plane wave travelling in the dielectric medium (a derivation is not required, but please include in the explanation the meaning of each variable in the above formula). [7 marks] d) Assuming the loss term above is zero, calculate the phase and group velocity within ˜ the dielectric, given that k˜ r . c [5 marks] e) What is anomalous dispersion? [3 marks] Page 5 Question 4. a) In the time dependent and static cases, the Magnetic Flux Density B, may be represented by the curl of a vector potential field given by, B A . Why is this so? Define the scalar potential for the static and time dependent cases, how do they differ? [6 marks] Consider two infinite conductor plates with charge density 0 c/m2 as shown. b) In the rest frame S0(x0,y0,z0), what is the magnetic flux density (B0) and electric field intensity (E0) between the plates. [5 marks] c) The moving frame S(x,y,z) has a velocity of v0 xˆ with respect to S0(x0,y0,z0) along the x axis as shown. Determine the magnetic flux density (B) and electric field intensity (E) between the plates in the moving frame, in terms of 0, the Lorentz contraction factor 0, v0, 0 and 0. (Remember you are dealing with vectors) [9 marks] (END OF PAPER) Page 6