Maxwell Equations - Electromagnetism and Radiation - Exam, Exams of Electromagnetism and Electromagnetic Fields Theory

Download Maxwell Equations - Electromagnetism and Radiation - Exam and more Exams Electromagnetism and Electromagnetic Fields Theory in PDF only on Docsity! The Handbook of Mathematics, Physics and Astronomy Data is provided KEELE UNIVERSITY EXAMINATIONS, 2011/12 Level III Tuesday 17th January 2012, 13.00-15.00 ASTROPHYSICS PHY-30004 ELECTROMAGNETISM AND RADIATION Candidates should attempt to answer THREE questions. A sheet of useful formulae and vector identities can be found on the last page. NOT TO BE REMOVED FROM THE EXAMINATION HALL PHY-30004 Page 1 of 7 1. An electric field in free space has the form E = sin(kx − ωt) ĵ + cos(kx − ωt) k̂ V m−1 , where the symbols have their usual meanings. (a) Show that this field is consistent with Gauss’s law in vacuum. [10] (b) Use Maxwell’s equations to calculate the associated time-dependent magnetic field. [20] (c) Show that the ratio of the E-field and B-field amplitudes is given by ω/k. [10] (d) Show that E and B are perpendicular. [10] (e) Give a definition of the Poynting vector in physical terms and calculate its numerical value for these electromagnetic fields. [25] (f) Calculate the vector force exerted by these fields on a 1 cm2 square of perfectly reflective material for each of the two sce- narios illustrated in the diagram below. [25] 45 o x y x y (ii)(i) /Cont’d PHY-30004 Page 2 of 7 4. (a) In an LIH medium, Ampère’s law can be written as ∇× B = µr µ0 J + µr µ0 ǫr ǫ0 ∂E ∂t , where the symbols have their usual meanings. i. Explain what physical conditions the acronym LIH refers to. [15] ii. If the medium is also a conductor with conductivity σ, then show that ∇ 2B = µr µ0 σ ∂B ∂t + µr µ0 ǫr ǫ0 ∂2B ∂t2 . [20] (b) The solutions to the equation in a(ii) may be waves of the form B = B0 exp[i(kx− ωt)] ĵ, where ω is an angular wave frequency and the magnitude of the wave-vector, k, is given by k2 = i µr µ0 σ ω + µr µ0 ǫr ǫ0 ω 2 . Describe the nature of this solution in two limiting cases: where σ ≪ ǫr ǫ0 ω and where σ ≫ ǫr ǫ0 ω. Your answer should include a discussion of any dependence of the wave amplitude on x and the wave velocity and wavelength compared to their vacuum values. [45] (c) A nuclear submarine communicates using radio waves of fre- quency 100 Hz. Justifying any assumptions that you make, es- timate to what depth the submarine must rise in order to com- municate with a surface station [For sea water you may assume that µr = 1, ǫr = 70 and σ = 5 S m −1.] [20] /Cont’d PHY-30004 Page 5 of 7 5. (a) Discuss whether Maxwell’s equations and electromagnetic fields are consistent with Special Relativity. Your discussion should include which quantities are invariant in different reference frames and which are not. [30] (b) Starting from Maxwell’s equations in vacuum, show that the electric field obeys a wave equation of the form ∇ 2E − 1 c2 ∂2E ∂t2 = 0 and identify c with a combination of physical constants. [20] (c) Explain how Maxwell’s equations are modified to treat the prop- agation of electromagnetic waves in a neutral, non-conducting, LIH medium and show that the wave speed is changed. [15] (d) Use the boundary conditions for electric fields at an interface between two media, and the example of electromagnetic waves normally incident on such an interface, to argue that the wave frequency is unchanged in a neutral, non-conducting medium. [15] (e) A Young’s double-slit interference experiment is carried out in (i) a vacuum and (ii) immersed in a non-conducting, non- magnetic, neutral fluid. For case (i), the first maximum in the interference pattern occurs at an angle of 10◦ to the incident light; in case (ii) the first maximum is shifted to an angle of 7◦. Explain this result and calculate the dielectric susceptibility, χe, of the fluid. [20] /Cont’d PHY-30004 Page 6 of 7 Electromagnetism formulae and vector identities Maxwell’s equations are ∇ · D = ρ ∇× E = − ∂B ∂t ∇ ·B = 0 ∇×H = J + ∂D ∂t where the symbols have their usual meanings. In LIH media D = ǫrǫ0E and B = µrµ0H. The electromagnetic potentials and the Lorenz gauge are defined by B = ∇×A E = − ∂A ∂t −∇V ∇ · A + 1 c2 ∂V ∂t = 0 Useful identities (where in these examples A is any vector field and V any scalar field) ∇× (∇×A) = −∇2A+∇(∇·A) ∇· (∇×A) = 0 ∇× (∇V ) = 0 ∮ A · dS = ∫ (∇ · A) d3r ∮ A · dl = ∫ (∇× A) · dS The Div and Curl operators in spherical polar coordinates (r, θ, φ) are given by ∇.A = 1 r2 ∂ ∂r (r2Ar) + 1 r sin θ ∂ ∂θ (sin θAθ) + 1 r sin θ ∂Aφ ∂φ ∇×A = 1 r2 sin θ ∣ ∣ ∣ ∣ ∣ ∣ ∣ r̂ rθ̂ r sin θφ̂ ∂ ∂r ∂ ∂θ ∂ ∂φ Ar rAθ r sin θAφ ∣ ∣ ∣ ∣ ∣ ∣ ∣ The elemental surface area and volume in spherical polar coordinates dS = r2 sin θ dθ dφ dV = r2 sin θ dr dθ dφ PHY-30004 Page 7 of 7