Electrodynamics (I): Homework 2 Due: April 2, 2015 Exercises in Griffiths 9.5(10% hand-in), 9.7(a)(b), 9.8, 9.11, 9.12, 9.13, 9.33, 9.34 Ex.1 10% hand-in Consider an infinitely long string with tension T . The string is align with the x-axis with a mass m attached to the string at x = 0. The direction of gravitational force is along −z direction. A travelling sinusoidal wave with wavevector k and angular frequency ω is incident from x = −∞, find the reflection coefficient (neglect effects of gravitation on waves). Ex.2 hand-in (a) 5% In the complex notation, the electric field for the most general electromagnetic wave ~ in vacuum can be written in the form Ẽ(~r, t) = (Aeiδ1 ê1 + Beiδ2 ê2 )ei(k·~r−ωt) , where A, B, δ1 and δ2 are real numbers, eˆ1 × ê2 = k̂, and eˆ1 · ê2 = ê1 · ~k = ê2 · ~k = 0. The corresponding h i electric field is Re Ẽ(~r, t) = E1 ê1 + E2 ê2 . Find the relation that E1 and E2 satisfy and show that it desribes an ellipse. Hence the general monochromatic EM wave exhibits elliptical polarization. (b) 5% Consider the superposition of two equal-amplitude, monochromatic, plane waves that propagate in opposite directions. Let the wave propagating along +z have left circular polarization and the wave propagating along −z have right circular polarization. Find the polarization of the resulting wave and the relation between the electric field and magnetic field. Repeat the above calculation when both waves are left circular polarized. Ex.3 hand-in (a) 5% The complex electric field of an electromagentic wave packet in vacuum is given by h i 1 Z 3 ~k) exp i(~k · ~r − ckt) . d k Ẽ ( Ẽ(~r, t) = ⊥ (2π)3/2 Find the total energy and total linear momentum of the wave packet in terms of the complex amplitude Ẽ⊥ (~k). (b) 5% For a Gaussian wavepacket, the complex amplitude Ẽ⊥ (~k) is given by 1 (~k − ~k0 )2 Ẽ⊥ (~k) = √ 3 exp − , ( aπ) a2 where a is a positive real number and ~k0 = (kx0 , ky0 , kz0 ) is a real vector. Find the total energy for general a and estimate the total momentum for this Gaussian packet when a → 0. 1

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# Electrodynamics (I): Homework 2 Due: April 2, 2015