ABSTRACT
For a simple lossless medium, derive the wave equations for E and H.
Obtain the wave Equations 1.18 and 1.19 subject to the Lorentz condition given by Equation 1.21.
Obtain Equation 1.27.
Show that for a time-harmonic case, Equation 1.22 reduces to Equation 1.50.
Obtain the expression for the time-harmonic retarded potential A for a current element of small length h located on the z-axis as shown in the figure.
Hence obtain the electric and magnetic fields at P. Assume that the current in the filament is I 0(A).
In the far zone, that is, for R ≫ λ, h ≪ λ, obtain E , H , E × H, and ⟨S⟩.
https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315272351/41040b79-9956-496a-ac34-683fa4364729/content/figu1.jpg"/>Lecture module-week 1 is devoted to reviewing Maxwell’s equations. In the undergraduate prerequisite course(s), considerable amount of time is spent in arriving at these equations based on experimental laws. In the process, vector calculus and coordinate systems are explained at length. The questions for the week make you revise the undergraduate background. You can look into the textbook you used in your undergraduate prerequisite course or search the internet in answering these two questions. The questions are:
Equations 1.1 through 1.4 are in differential form. Write down the corresponding equations in the integral form.
The integral form of Equation 1.1, Faraday’s law, allows you to compute the voltage induced (emf) in a circuit. There are two components to it: (i) transformer emf and (ii) motional emf. Give an example where only (i) is induced; give an example where only (ii) is induced and give a third example where both (i) and (ii) are induced. In the third case, if your interest is in computing 590the total induced voltage and not in separating them into the two components, is there a simpler way of computing the total induced voltage? Illustrate the simpler way through an example.
This question is to make you think about the concept of retarded potentials. Equations 1.22 and 1.24 are useful in answering this question. Note that Equation 1.22 is written when the source is a volume current element. For a filamentary current source, you will replace J dV by I dL.
Suppose the source is a differential filamentary current element located at the origin along the y-axis, where I d L → = 2 t y ^ d y . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315272351/41040b79-9956-496a-ac34-683fa4364729/content/equ15_1.jpg"/> Obtain the differential vector potential d A → P https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315272351/41040b79-9956-496a-ac34-683fa4364729/content/eqi15_1.jpg"/> at the point P (5, 10, 15).
Note the following. You can use a bold letter or a letter with an arrow on the top to denote a vector. The hat on the top of y denotes a unit vector in the y-direction in the Cartesian system of coordinates (rectangular coordinate system).
Consider a cylindrical capacitor of length ℓ. The radius of the inner cylinder is a and the radius of the outer cylinder is b. The dielectric constant of the dielectric inside the capacitor varies as (K is a constant) ε r ρ = K ρ 2 . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315272351/41040b79-9956-496a-ac34-683fa4364729/content/equ15_2.jpg"/> Show that the conduction current I(t) in the wire is the same as the displacement current in the capacitor. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315272351/41040b79-9956-496a-ac34-683fa4364729/content/figu2.tif"/>
A parallel plate capacitor is of cross-sectional area A and is filled with a dielectric material whose permittivity varies linearly from ε = ε1 at one plate (y = 0) to ε = ε2 at the other plate (y = d). Neglecting fringing effects, determine the capacitance. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315272351/41040b79-9956-496a-ac34-683fa4364729/content/figu3.tif"/>
Show that the displacement current between two concentric cylindrical conducting shells of radii r 1 and r 2, r 2 > r 1 is exactly the same as the conduction current in the external circuit. The applied voltage is V = V 0 sin ωt. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315272351/41040b79-9956-496a-ac34-683fa4364729/content/figu4.tif"/>
Derive (2.42) through (2.44), based on Section 1.6.
Determine the vector H field at a general point in free space due to a DC current I A in an infinitely long filament lying on the z-axis. Use Ampere’s law and symmetry arguments.
Suppose that an infinite sheet of current of surface current density x ^ K A / m https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315272351/41040b79-9956-496a-ac34-683fa4364729/content/eqi15_2.jpg"/> lying on the xy plane. Determine H at a general point P(x, y, z). Use Ampere’s law and symmetry arguments. Show that the boundary condition (1.17) is satisfied by your results.
Equation 1A.14 is the well-known divergence theorem. Less well known are some more theorems that can be derived from the divergence theorem. These are (1A.68) and (1A.69). Prove them
Equation 1A.66 is the well-known Stoke’s theorem. Less well known are some more theorems that can be derived from the Stoke’s theorem. One of them is (1A.70). Prove it.
Derive wave equation for electroquasistatics and magnetoquasistatics discussed in Section 1.5. Assuming the medium is free space and the fields vary with only z coordinate, obtain their form.
Derive an expression for ψ, the angle between the position vectors r and r ′. Express it in terms the of their spherical coordinates.
