Exercise 19

As with all of the exercises you will complete in this course, you will be expected to turn your *.nb file in to the course TA before the next lecture.  Please do so by e-mailing your completed exercise file to Lan Yu (ece329.uiuc@gmail.com), make sure you name the file using the following notation: “Exn_Last_First.nb”, where n is the Exercise number (in this case, 19), and First and Last would be your first and last name.  So my submission to Lan for this exercise would be titled “Ex19_Wasserman_Dan.nb”.  You will receive full credit for your exercise if it turned in before the next lecture (Lecture n+1).  No late submissions will be accepted.

Objective/Overview:

In today’s Lecture (19) we worked with d’Alembert waveforms of TEM plane waves and more importantly, discussed the origin of these plane waves.It turns out that our plane waves are generated by time-varying surface currents.  these surface currents generate H fields on either side of the current, but we can use (OverVector[H])∝(Subscript[(OverVector[J]), s])/2 only right at the surface...if the field everywhere above and below was proportional to current at any instant in time, then we would be transporting information and energy faster than the speed of light.  So we have to include a time-delay to determine the H field away from the surface current.  In today’s exercise we will start with surface currents and use these to understand the propagation of TEM plane waves originating at a surface-current source

Surface Current Sources of TEM Plane Waves

Let’s see if we can start with an arbitrary surface current and determine the H and E everywhere in space from this current.
If our surface current is on the xy-axis, and described by (OverVector[J])=(Subscript[J, s])·t·rect((t/τ))(OverHat[y]), where (Subscript[J, s])=4A/m and τ=4μs, let’s find and plot E and H assuming our current is surrounded by vacuum.
First we write out the expression for our surface current.  We will treat all of our vector as scalars and figure out directions at the end:

js[jso_,tau_,t_]:=jso*t*(UnitStep[t+tau/2]-UnitStep[t-tau/2])

Let’s plot this:

Plot[js[4,4,t],{t,-5,5}]

OK, that looks right.
Now we should be able to find H from our Js.  Just remember that because we have a surface current, we will have a discontinuity in H at z=0, so we wil have to write H as a piecewise function, taking into account BOTH the different propagation directions and the different signs.

h1[jso_,tau_,t_,z_]:=0.5*Piecewise[{{js[jso,tau,t-z/300],z>0},{-js[jso,tau,t+z/300],z<0}}]

We can now plot our H field as a function of time and position.  Let’s first plot as a function of position for increasing time:

Plot[h1[4,4,t,z]/.{t->3},{z,-2400,2400}, PlotRange->{-4,4}]

Step through time: t=1, 2, 3, etc in the above plot.  Make sure you understand how the wave evolves in time.
Now pick two symmetric positions, let’s say z=±600m.  Plot h1 at these positions as a function of time: 
Hint: use the command “Table[Plot[h1[...” and let z take two different values...+/-600

Table[Plot[h1[

Now we are ready to add our Electric Field.  Remember, boundary conditions tell us that the Electric field is constant across z=0, and because (OverVector[E])×(OverVector[H]) gives the direction of propagation, we see that (OverVector[E])∝-(OverHat[y]).

e1[jso_,tau_,t_,z_]:=-0.5*120*Pi*Piecewise[{{js[jso,tau,t-z/300],z>0},{js[jso,tau,t+z/300],z<0}}]

Plot[e1[4,4,t,z]/.{t->3},{z,-2400,2400},PlotRange->{-1500,1500}]

Now we can plot E and H at the same time.  Let’s put it all together!

plote1=Animate[Plot[e1[4,4,t,z],{z,-2400,2400},PlotRange->{-1500,1500}],{t,-5,5}];

ploth1=Animate[Plot[h1[4,4,t,z],{z,-2400,2400},PlotRange->{-4,4}],{t,-5,5}];

Table[{plote1,ploth1}]

Note that your E field is symmetric about the z axis, and your H field is antisymmetric (as would be expected, since H has opposite sign on opposite side).

Your Turn

Now suppose you have a surface current (OverVector[J])=(Subscript[J, s])(rect(((t-1)/τ))-2rect(((t+1)/τ)))(OverHat[x]) on the xz plane. With (Subscript[J, s])=2A/m and τ=2μs.  Follow our example above and calculate and plot (OverVector[J])(t), (OverVector[H])(z,t), and (OverVector[E])(z,t).  Make sure you note the directions of the fields.