Problems for Edward F. Redish |
An electromagnetic plane wave moving in the z direction can be written as (the real part of)
From Maxwell's equations in free space, we can show that the amplitude vector, E_{0}, has to be perpendicular to the direction of propagation and therefore lie in the the x-y plane. We could therefore treat the amplitude vector as a two-dimensional vector in a complex linear space:
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(a) If both E_{0x} and E_{0y} are real numbers (with dimensions of force/charge), find the electric field vector at the origin (i.e., (x,y,z)=(0,0,0)) as a function of time. What is it's magnitude and direction as a function of time? (Hint: Remember that the physical electric field is the real part of the complex expression.)
(b) Suppose both E_{0x} and E_{0y} are complex numbers with the same phase, that is,
where A_{0x} and A_{0y} are both real (but not necessarily equal) and the φ in both terms is real (and they are both the same number). How does your result from part (a) change?
(c) Now suppose that instead of having the same phase, they are 90^{o} out of phase. That is, suppose that:
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Find the electric field vector at the origin as a function of time. Describe how it is changing.
(d) We can consider two distinct bases for the linear space of these 2-D E fields, {|e_{1}>, |e_{2}>} and {|+>, |->}, where:
We can express any E-field vector in one of the two ways:
Explain what each of the coordinates E_{1}, E_{2}, E_{+}, E_{-} mean and express {E_{+},E_{-}} in term of {E_{1},E_{2}}.
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Last revision 28. November, 2005.