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Unformatted text preview: Electromagnetic Theory II
Problem Set 11
Due: 11 April 2011 43. The electric ﬁeld of a plane wave ei · is perpendicular to the propagation direction k.
The polarization vector is by deﬁnition a unit vector parallel to the electric ﬁeld. We have
found it useful to describe the ﬁelds with complex vectors whose real parts are the physical
ﬁelds. In this description polarization vectors can be complex when they describe elliptic
polarization, in which case we adopt the normalization ∗ · = 1. Complex vectors belong to
a three dimensional complex vector space, and it is convenient to pick an orthonormal basis.
Let n = k be the unit vector parallel to k. Then choose e3 = n, e1 = , and e2 = n × ∗ .
¡ a) Show that the three ea are othonormal in the sense that e∗ · eb = δab , and prove the
completeness relation 3=1 ei ej ∗ = δij . Note that the completeness relation can be
2 ei ej ∗ = δij − ei ej ∗ = δij − ni nj
a=1 which is useful to sum cross sections over polarizations, p ol i j∗ = δij − ni nj . b) We can expand any real unit vector in three space v = vx x + vy y + vz z in the new basis
v = a Va ea . Show that a |Va | = 1
c) Prove the identity
|r · n|2 + |r · |2 + |r · (n × )|2 = 1
where r is the radial unit vector.
44. J, Problem 10.1. Hint: apply the results of problem 40 to the result in Eq. (10.14).
45. J, Problem 10.2. Again it is suﬃcient to work with Eq. (10.14).
46. J, Problem 10.3. 1 ...
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This note was uploaded on 01/08/2012 for the course PHY 6346 taught by Professor Staff during the Spring '08 term at University of Florida.
- Spring '08