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Lecture2 - Lecture 2 Notes 95.658 Spring 2012...

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Lecture 2 Notes, 95.658, Spring 2012, Electromagnetic Theory II Dr. Christopher S. Baird, UMass Lowell 1. Dispersion Introduction - An electromagnetic wave with an arbitrary wave-shape can be thought of as a superposition of single-frequency plane waves. - If the dielectric material responds in the exact same way to plane waves of any frequency, than each component of the wave-shape will travel at the same speed, and the overall wave-shape will be preserved. - If the dielectric material responds differently to plane waves of different frequencies (i.e. its dielectric constant is frequency dependent), then the superposed components of the wave-shape will travel at different speeds. The wave-shape will change in time as it propagates through the material and in general will spread out. - This wave spreading is known as dispersion. - We must explicitly describe the behavior of the frequency-dependent permittivity ε ( ω ), or in other words, the dielectric constant ε r ( ω ) = ε ( ω )/ ε 0 , before we can describe the dispersion in a detailed way. - An accurate prediction of a material's permittivity requires quantum mechanics, but we can find a good approximation by building a classical model. 2. Classical Harmonic Model - Remember that the dielectric constant describes a material's response to an applied electric field. - The applied field induces dipoles in the material which then create their own electric fields that add to the original field. - So if we find the dipole moment induced in a single atom by an applied field, we can sum over all atoms and find the dielectric constant. - For simplicity, we will deal only with non-magnetic dielectric materials, so that the magnetic permeability equals the permeability of free space. We also ignore magnetic forces. - Consider a single electron bound to an atomic nucleus pushed by a wave that passes by. Assume the atomic nucleus is so much heavier than the electron that it stays fixed. - Newton's Law states: F = m a where F is the total force, m is the electron's mass and a is the electron's acceleration. F wave F binding = m d 2 x dt 2 e E k x = m d 2 x dt 2 where k is the spring constant of the harmonic binding force. For a harmonic system with a single mass the spring constant is k = m ω 0 2 where ω 0 is the resonant frequency of the spring (which is here caused by the binding force in the atom). + - e F wave F binding
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e E m 0 2 x = m d 2 x dt 2 - In its current form, this equations describes an electron that would oscillate forever. In reality, the oscillating electron interacts with other electrons, looses energy in the process, and slows down. - Instead of going into the details of the damping, we can sum up the process in a phenomenological damping parameter γ : e E m 0 2 x = m d 2 x dt 2 m d x dt - Let us investigate the material response to a plane wave at a single frequency ω , then we can always super-impose plane waves for more complex situations: e E x e i t m 0 2 x = m d 2 x dt 2 m d x dt - We try a solution of the form x = x 0 e i ω t :
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