Time left 0:01:01Question 11Not yet answeredMarked out of 15.00Flag questionTipsQuestion textThe electric and magnetic fields of a short dipole (length L << λ) that is placed at the origin and oriented parallel to the z axis of the standard spherical coordinate system can be calculated in the general case at the point r,θ,φ from the following formulaeE⃗ =Erur+Eθuθ𝐸→=𝐸𝑟𝑢𝑟+𝐸𝜃𝑢𝜃 whereEr=j2ωμILcosθe−jkr4πr(1jkr+1(jkr)2)𝐸𝑟=𝑗2𝜔𝜇𝐼𝐿cos⁡𝜃𝑒−𝑗𝑘𝑟4𝜋𝑟(1𝑗𝑘𝑟+1(𝑗𝑘𝑟)2)Eθ=jωμILsinθe−jkr4πr(1+1jkr+1(jkr)2)𝐸𝜃=𝑗𝜔𝜇𝐼𝐿sin⁡𝜃𝑒−𝑗𝑘𝑟4𝜋𝑟(1+1𝑗𝑘𝑟+1(𝑗𝑘𝑟)2) andH⃗ =Hϕuϕ=uϕjkLIsinθe−jkr4πr(1+1jkr)𝐻→=𝐻𝜙𝑢𝜙=𝑢𝜙𝑗𝑘𝐿𝐼sin⁡𝜃𝑒−𝑗𝑘𝑟4𝜋𝑟(1+1𝑗𝑘𝑟)ω is the angular frequency, I is the amplitude of the electric current in the dipole, L is the length of the dipole, and k = 2π/λ is the wave number.(a) Simplify the expression for the wave impedance of a short dipole η=|E⃗ |/|H⃗ |𝜂=|𝐸→|/|𝐻→| as much as possible when θ = 90°. (Keep in mind part b)(b) Using the software of your choice, plot the wave impedance η as a function of the distance-to-wavelength ratio r/λ along the axis perpendicular to the antenna (θ = 90°). Choose, e.g., 0.05≤r/λ≤3 and 0 Ω≤η≤1000 Ω.(c) Mark on the plot i) the level where η = η0 =377 Ω, ii) the distance r/λ = 1/(2π), and iii) the distance r/λ at which η = 0.997η0.(d) What is the reactance of a short dipole like? Which field (E or H) dominates in the distance range r/λ = 1/(2π)? How is this related to the reactance of a short dipole?(e) What distance can be considered as the outer boundary of the reactive near fields of a short dipole?(f) What is the distance of the far field region of a short dipole?(g) How would you assume that the wave impedance curve changes for a small loop antenna? What is the reactance of a small loop, inductive or capacitive?

Field and flux of a magnetic dipole. The magnetic field at a point M(r, θ) distant from a magnetic dipole m is expressed as B(M) = − µ0 4π ∇ m · r r 3  This formula is identical to that of the electric field produced by an electric dipole moment m, after changing µ0 into 1/0. q m i z r ur uq O O’ M a R’ We consider a small planar and circular loop of center O, radius a, carrying a current i and we use a spherical coordinate system (r, θ, ϕ) with the z-axis oriented along the magnetic dipole moment: m = muz. The expression of the gradient in spherical coordinates is as follow: ∇φ = ∂φ ∂r ur + 1 r ∂φ ∂θ uθ + 1 r sin θ ∂φ ∂ϕuϕ a. Express the corresponding magnetic dipole moment as a function of a and i.

The centre of a half-wave dipole is placed at the origin of the standard spherical coordinate system with the wire parallel to the z axis. This centre-fed dipole is in resonance at 0.9 GHz and the total radiated power is 200 W.(a) Calculate the effective height vector in the xy plane (θ = 90°).(b) Find the electric field vector (peak value) radiated by the dipole in the far-field region.(c) A second similar dipole is also placed in the xy plane parallel to the first at a distance of 1500 m. Neglecting losses and assuming the free-space condition, calculate the voltage U (peak value) induced in the dipole, i.e. the voltage in the equivalent receiving circuit.(d) Based on the voltage U, calculate the maximum rms power available at the matched load

Draw a diagram of a dipole at height h above a flat ground plane,and orientated at angle α with the ground plane. Set the origin of yourvector space to be the point on the ground directly beneath the centre of thedipole, and use the method of images to find an expression for the E-field ofthe dipole with the ground plane.To simplify this problem, you may assume you know the angles, θ, in theexpression for the dipole field for both the dipole and its image.1

Derive the expression for the electric field of a dipole at a point on the equatorial plane of the dipole.

At a point on the axis of an electric dipole