Please solve the problems provided. Make sure to include all work, steps and details leading to the answer

Please solve the problems provided. Make sure to include all work, steps and details leading to the answer









ECE 407 – ELECTROMAGNETIC COMPATIBILITY ECE Advanced Electromagnetic Fields and Waves I 1. A dipole antenna lies on the z-axis between −?/2 and ?/2 in free space. Let’s consider a triangular distribution for the current over the dipole: ?(?) = ?0 (1 − | ? ?/2 |) a) Derive the far field radiated by the dipole. b) Derive the radiation resistance and plot the radiation resistance as a function of the length for 0 < ?/2 ≤ 1. the involved integral for this question should be calculated numerically. c) assume that ?/2 ≪ 1 and determine an approximate formula for the radiation resistance. compare this to the result for a linear dipole antenna from chapter 2 of the notes. compare the numerical value to that obtained in part (b) for ? = λ/10. 2. consider a hertzian dipole at the origin of a cartesian system and oriented along a generic direction �̂�. (a) derive the radiated field. (b) derive the equivalent currents over the surface of a sphere centered with the dipole and with a radius of 5λ. 3. consider a sphere with center at the origin of a cartesian system and with radius 5 λ. the following electric (?) and ( �⃗⃗⃗�) magnetic current distributions are provided over the surface of the sphere: ? = ?0 sin ? ? �⃗⃗⃗� = ?0 ζ sin ? �̂� with ?0 a constant and ζ the characteristic impedance in free space. (a) (15 pts) derive the field radiated by such a distribution in a generic point in free space within and outside the sphere. hint: link these currents to the point (b) of ex. 2 for a dipole along the z-axis. (b) the far-field distance associated to such distribution. 2="" ≤="" 1.="" the="" involved="" integral="" for="" this="" question="" should="" be="" calculated="" numerically.="" c)="" assume="" that="" 2="" ≪="" 1="" and="" determine="" an="" approximate="" formula="" for="" the="" radiation="" resistance.="" compare="" this="" to="" the="" result="" for="" a="" linear="" dipole="" antenna="" from="" chapter="" 2="" of="" the="" notes.="" compare="" the="" numerical="" value="" to="" that="" obtained="" in="" part="" (b)="" for="" =="" λ/10.="" 2.="" consider="" a="" hertzian="" dipole="" at="" the="" origin="" of="" a="" cartesian="" system="" and="" oriented="" along="" a="" generic="" direction="" �̂�.="" (a)="" derive="" the="" radiated="" field.="" (b)="" derive="" the="" equivalent="" currents="" over="" the="" surface="" of="" a="" sphere="" centered="" with="" the="" dipole="" and="" with="" a="" radius="" of="" 5λ.="" 3.="" consider="" a="" sphere="" with="" center="" at="" the="" origin="" of="" a="" cartesian="" system="" and="" with="" radius="" 5="" λ.="" the="" following="" electric="" (?)="" and="" (="" �⃗⃗⃗�)="" magnetic="" current="" distributions="" are="" provided="" over="" the="" surface="" of="" the="" sphere:="" =="" 0="" sin="" �⃗⃗⃗�="?0" ζ="" sin="" �̂�="" with="" 0="" a="" constant="" and="" ζ="" the="" characteristic="" impedance="" in="" free="" space.="" (a)="" (15="" pts)="" derive="" the="" field="" radiated="" by="" such="" a="" distribution="" in="" a="" generic="" point="" in="" free="" space="" within="" and="" outside="" the="" sphere.="" hint:="" link="" these="" currents="" to="" the="" point="" (b)="" of="" ex.="" 2="" for="" a="" dipole="" along="" the="" z-axis.="" (b)="" the="" far-field="" distance="" associated="" to="" such="">