Physics 361 Problem set 4 due Wednesday November 4, 2009 P4.1 surface plasmon vs thin-ﬁlm plasmon Charge-oscillation modes occur at the surface of an electron gas. The best known one is for charge oscillations of wavevector q along the surface of an electron gas with density n in which the ﬁelds die oﬀ exponentially with distance on either side of the interface: q 2 = ( ω/c ) 2 ( ω 2 p-ω 2 ) / ( ω 2 p-2 ω 2 ). Consider a thin layer of electron gas with n s electrons per unit area. As usual, there is a compensating uniform background charge. Suppose that the electrons are displaced from their initial uniform state by an amount u ( x ) = u0 ˆ x sin qx . This produces a nonuniform charge density proﬁle n s ( x ) a) Find the charge density proﬁle n s ( x ). b) Find the electrostatic potential φ ( x ) resulting from this charge density. For this it is helpful to know the electrostatic potential
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