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09ffinal - Electromagnetic Theory I Final Examination 18...

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Electromagnetic Theory I Final Examination 18 December 2009, 12:30-2:30 pm Instructions: Answer the following 10 questions, each of which is worth 10 points. Explain your reasoning in each case. Use SI units throughout. Points will be deducted for gaps in logic as well as for errors in calculation. Extra points will be deducted if an incorrect answer is presented with incorrect units! A sheet of useful formulas is provided at the end of the exam booklet. 1. (10 points) Starting with Maxwell’s equations in vacuum (i.e. = 0 , μ = μ 0 ), prove that the electric and magnetic fields satisfy wave equations of the form -∇ 2 + 1 v 2 2 ∂t 2 ψ = J. Be sure to derive the wave speed v in terms of the parameters 0 , μ 0 , and to also derive the source term on the right side for both electric and magnetic fields. 1
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2. (10 points) The electric dipole moment induced by a weak electric field always aligns with the field. In contrast, the magnetic dipole moment induced by a weak magnetic field can, depending on the substance, be aligned with the field (paramagnetic) or opposite to the field (diamagnetic). Explain this qualitative difference in electric and magnetic response in terms of the different ways in which the electric and magnetic fields contribute to the dynamics of a charged particle. (Hint: Consider the different field dependence of the electric dipole moment d e and magnetic dipole moment d m of a particle: d e = q r , d m = q 2 m r × ( p - q A ) where B = ∇× A , r is the particle’s position coordinate and p is its canonical momentum.) 2
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