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Problem Set 8 : using Huygens equivalence principle

Typology: Exercises

2023/2024

1 / 2

Download Problem Set 8 : Equivalence principle and more Exercises Guiding Electromagnetic Systems in PDF only on Docsity! Problem Set #8 1. A slab of current density in the region −∞ < x < ∞ , −∞ < y < ∞ , –Δ < z < Δ has J = ŷ J0 . (a) Use the Green’s function e− jkz z 2 jkz to find the magnetic vector potential Ay as a function of z in the z > Δ region. What is kz ? (b) Find the Hx field as a function of z in the z > Δ region. 2. Find equivalent electric and magnetic surface currents (Note #21) located on a sphere of radius a, centered at the origin, that if radiating in free space reproduce the fields of a Hertzian dipole at the origin. (Hint: Define these directly in spherical coordinates.) 3. The fields of a thin, linear dipole are often obtained by approximating the equivalent electric current density by the “sinusoidal triangle” function considered in Problem Set #7, Problem 4: J (x, y, z) = ẑ I0 sin kh − k z( )δ (x)δ (y)p(z;−h,h) Using the result of Problem 4 on Problem Set #7 to assist you, show that the ẑ component of the electric field of this source can be expressed without approximation as Ez (x, y, z) = − jηI0 e− jkR1 4πR1 + e− jkR2 4πR2 − 2cos(kh) e − jkr 4πr ⎧ ⎨ ⎩ ⎫ ⎬ ⎭ What are R1 and R2 in this expression? Hint: Use the fact that