EE6506-Assignment 1 Solved

1.   For a volume V with a surface S and scalar functions ψ, φ, prove:

       dv        : Green’s first identity

      (b)  φ∂n − ψ∂ )ds = RV(φ∇2ψ − ψ∇2φ)dv : Green’s second identity

2.   Given that the instantaneous electric field inside a source-free, homogeneous, isotropic, and linear medium is given by E = [α(x + y)xˆ + β(x − y)yˆ] sin(ωt), find a relation between α, β.

3.    An electric line source of infinite length and constant current along the z axis radiates in free space, and at large distances from the source the expression for the magnetic field is as follows: H~ = φˆH0 exp(−jβ0ρ)/√ρ, (H0 is a constant). Find the corresponding electric field.

4.   (MATLAB) Construct a hill using a 2D gaussian function, e.g. z = exp(−(x − 5)2 − 4(y − 3)2). Write a function that creates an outward normal to the hill’s surface given a point on it as input. Now, construct a contour around the hill whose height is half the maximum height. At equispaced points on this contour, plot the outward normals. In the main body of the submission, your plot must show the hill and the chosen contour as well. Hint: use the quiver3 function. You must include your code (with comments) in the submission as an appendix, as well as a stand-alone .m file.

5.   (MATLAB) Assume the time harmonic form of Maxwell’s equations (i.e. time dependence is as exp(jωt)). Write a code that takes as input an electric field and returns the magnetic field. Note that you must write a symbolic code. For e.g. your code should be able to take an electric field of the following form as input: ~E = (x sin(z), y2, z3x). Hint: Look up the symbolic toolbox. This problem will prepare you to learn a very powerful way of doing math and scientific computing. You must include the output of the code to the above example in the main body of the submission. You must include your code (with comments) in the submission as an appendix, as well as a stand-alone .m file.

6.   Give a short proof of the uniqueness theorem, being sure to point out why the medium needs to be lossy.

7.   Derive a continuity relation for the normal components of D~ at an interface between two media in the presence of a surface charge.

8.   Derive (but do not solve) the form of the differential equation obeyed by a time-harmonic wave travelling along the z axis of a waveguide at a point away from the exciting current. The cross-section of the waveguide is not a function of the z co-ordinate, but an arbitrary e(x, y). Hint: What is the logical and simplest form of the z-dependence of the field? Make simplifying assumptions like the choice of polarization, etc., and state them.