# Demo04Meta - ECE 3030: Electromagnetic Fields and Waves...

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ECE 3030: Electromagnetic Fields and Waves Fall 2009 DEMO 4 INSTRUCTOR NOTES: Meta Materials and Magnetic Fields Reminder: Prelim next week No Homework Set 4. Date: Tuesday 9/22 1. Problem 5.13 Determine the eﬀective permittivity of the nano-dot meta-material shown in Figure 1. Assume the dots are small enough and 1 ε 2 Figure 1: Nano-dot meta-material. widely enough spaced that the ﬁeld from one dot does not interact with the ﬁeld of any other dot. First consider ﬁrst just one nano-dot imbedded in the background dielectric with an applied uniform ﬁeld as shown in Figure 2. x z 2 z E E ˆ 0 = a θ Figure 2: A nano-dot of a meta-material. The distortion in the applied ﬁeld caused by the embedded sphere is not shown. a. Find trial solutions, Φ in ( ~ r) and Φ out ( ~ r), for the potentials inside and outside the dielectric sphere. (Hint: Φ out ( ~ r ) must have a term that has the same form as a dipole potential). b. Write down all the boundary conditions (at least as many as the num- ber of unknown constants in your answer to part (a) above) relevant to solving for the potentials, Φ in ( ~ r) and Φ out ( ~ r). c. Find all the unknown constants in your solutions in part (a) above by using all the boundary conditions in part (b) above. d Compare the dipole-like term in your solution for Φ out ( ~ r) to that of a point-charge dipole potential (see Section 3.5) and from that comparison ﬁgure out the dipole-moment ~ p (small ”p”) of the polarized dielectric sphere. Make sure you get the correct units. (Hint: the dipole moment CORNELL UNIVERSITY c ± WES SWARTZ (09/8/12; Rev.09/9/23) 4–1

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ECE 3030: Electromagnetic Fields and Waves Fall 2009 DEMO 4 INSTRUCTOR NOTES: Meta Materials and Magnetic Fields must be proportional to the applied E-ﬁeld magnitude). This dipole mo- ment has been ”induced” in the dielectric sphere due to the external E-ﬁeld. Now come back to the medium made up of nano-sized dielectric spheres
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## This note was uploaded on 01/16/2010 for the course ECE 3030 at Cornell University (Engineering School).

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Demo04Meta - ECE 3030: Electromagnetic Fields and Waves...

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