A number N of plane waves are travelling parallel; the waves share a common wavevectork and common direction of electric field vector along unit vector û, but each wave has itsown distinct amplitude Eor and phase difference εr. Therefore the ith wave (for 1 ≤ i ≤N)has electric field given byE₁ = Eoiû Re [exp j(kr - wt + 6;)] .-From this, prove that the total intensity I net including interference is given byInet=| Ec|2200"where the complex amplitude Ę is defined byNE₁ = Eoi exp(jdi),and no is the impedance of free space.i=1

Answered: A number N of plane waves are…

Principles of Physics: A Calculus-Based Text

5th Edition

ISBN:9781133104261

Author:Raymond A. Serway, John W. Jewett

Publisher:Raymond A. Serway, John W. Jewett

Chapter24: Electromagnetic Waves

Section: Chapter Questions

Problem 28P

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Problem 1. In class we will consider the sum of the electric field of two plane waves, u:(z.t) and uz(z.t), both traveling in the positive z direction with slightly different frequencies and propagation constants and with equal amplitudes. The electric fields of both plane waves are oriented in the y direction, which means that they have the same "polarization". The time and space variation of the two electric fields are given by: u: (2.1) = cos(@t - kz) uz(2.1) = cos ([o + Ao]t - [k + Ak]2) We will show that the sum of these two waves produce an intensity envelope in time and space that modulates the carrier frequency of m + Ao/2. The velocity of the envelope is called the group velocity. Assume Ao is << than o. For this problem, consider the sum of two slightly different waves with similar electric fields: v.(z) = sin(ot - kz) vz(z1) = sin([o + Ao]t - [k + Ak]z) Derive analytically an expression for a) the group and b) the phase velocity for the sum of the two fields. HINT: sinfa) +…

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