ATMO/ECE 489/589  Final Exam
May 9, 2017

Everyone should answer Question #1 (25 pts)
ATMO/ECE 489 students should then answer 3 of the remaining questions (20 pts each)
ATMO/ECE 589 students should answer any 4 of the remaining questions (20 pts each)

1.  Please answer any 5 of the following short answer questions (5 pts each)

(a)   Satellite observations indicate a global lightning flashing rate of about 45 flashes/second.  What would the corresponding global flash density (flashes/(km2 year)) be?

(b)   We might expect to measure an electric field of 100 to 300 V/m at the ground during fair weather.  What if it were foggy outside, would you expect to measure a higher, a lower, or the same value of the electric field.  Explain.

(c)   Despite a sizable increase in population, the number of people killed by lightning every year in the United States has decreased from more than 400 in the early 1900s to less than 30 people per year at the present time (R.L. Holle, "Some Aspects of Global Lightning Impacts," 2015 Amer. Meteorol. Soc. Meeting, Phoenix).  In addition to better hazard warning and safety education, automobiles play an important role in the decrease in lightning deaths.  Why do you think this is so?

(d)   To calculate the average score on an exam in a class like ours I would sum up the exam scores and divide by the number of students, i.e. I'd compute the arithmetic mean.  With lightning measurements such as peak return stroke current, the geometric mean is often given instead of the arithmetic mean.  How do you compute the geometric mean?  Under what circumstances might the geometric mean better characterize a group of measurements than the arithmetic mean?

(e)   List and briefly discuss the sources of atmospheric ionization present in (i) the lowest meter of the atmosphere, in (ii) the first 10s and 100s of meters above the ground, and (iii) between 1 and 10 km altitude.   Would the same list apply out over the ocean?

(f)   We generally assume that the electric field at ground level is perpendicular to the earth's surface.  Why is that the case?  Explain also why, even though the earth is roughly spherical, we generally write the E field at the ground as having a z-component (Ez) rather than a radial component (Er).

(g)   The transmission line model (TLM) is widely used to estimate peak return stroke current and current derivative values from distant measurements of electric and magnetic radiation fields.  What is the actual TLM  relationship between peak current and peak field (or peak current derivative and peak field derivative value)?  What assumptions are made when deriving the transmission line model equations.

(h)   I don't remember having mentioned dry lightning, ribbon lightning, bead lightning, heat lightning, or hot lightning in class this semester.  What is meant by these terms or descriptions?  Are these really different types of lightning?  (this was covered in Spring 2015, on  Feb. 27 as a matter of fact).

(i)    People standing at points A and B in the figure below are about the same distance from where lightning has struck the ground.  Would the thunder they hear be about the same or different?  If different, in what way(s)?
 

2.  Aircraft are struck by lightning fairly frequently.  The discharge is often triggered by the airplane itself and there have been several field campaigns that sought to better determine the conditions that could lead to a lightning trigger and to evaluate whether aircraft were adequately protected from a lightning strike.  You'll find a supplementary reading section on "Lightning interactions with aircraft" in this semester's lecture notes.

The aircraft in those research studies made measurements of the surrounding thunderstorm electric field.  Field mills are mounted at multiple locations on the aircraft body and a complex analysis procedure is used to determine the 3-dimensional field surrounding the aircraft and also to account for any charge that might have built up on the aircraft.  The locations of 5 field mills on the Convair CV580 and the Transall C160 are shown below

This question will make use of a much simpler geometry, a conducting sphere, and will assume the sphere is placed in a uniform, vertically oriented, ambient electric field (the aircraft studies generally assume that the field is uniform but of unknown orientation).

In our Thursday, Feb. 2 class, we derived the potential function, Φ(r,θ) in the space surrounding a sphere


The problem geometry is shown above.  The potential function that we ended up with is shown below


One of the boundary conditions that we used when working out the problem was that the potential was constant on the surface of the sphere, that is the case above when r = a.  Now we will imagine that the sphere is charged.  The potential function for a point charge is

We'll add this expression to the equation above for the uncharged sphere and end up with an expression for a charged sphere in a uniform field (here's a reference that convinced me this is a valid approach)

(the Φo term was dropped in this expression).  Note that Φ is constant on the surface of the sphere in this case also (r = a in the expression above).

This Final Exam question has three parts:

(a)  The charge on the sphere is spread out over the surface of the sphere.  Using the expression above show that the surface charge density on the surface of the sphere is



(b) Integrate this expression for surface charge density over the surface of the sphere

(c) Imagine you were able to measure the electric field at the top and bottom surfaces of the sphere (just as field mills are able to measure the electric field at various locations on an aircraft).  Show how you could use the measurements of Etop and Ebottom to determine both the ambient field, Eo, and the total charge, q, on the sphere.

3.  A negatively charged, downward propagating stepped leader is approaching flat ground.  About how high above the ground will the tip of the leader be at the moment upward connecting discharges are initiated at the ground and begin to move upward to intercept the leader?  You can assume that a peak current of 45 kA is measured in the ensuing return stroke once attachment between the leader and one of the upward connecting discharges is made.

4.    The point charge Q in the figure below is positioned a distance H above the ground (which you should assume is a conducting surface).  The electric field is being measured a distance D away.

(a)   Derive an expression for the electric field as a function of H/D.

(b)   Show that the electric field reaches a maximum when H/D = 1/√2

5.    Orthogonal components of the magnetic field have been measured at two locations in a network of magnetic direction finders.  Station #1 has arbitrarily been placed at the origin of the x-y coordinate system.  Station #2 is located to the northeast of Station #1 as shown below.  You can ignore any attenuation of the field amplitudes by propagation and can assume the ground is flat.



(a)  Determine the bearing vector from each station to the lightning strike point.

(b)  Determine the x, y coordinates of the strike point

(c)  What is the range normalized amplitude (100 km) of the B field (magnetic field)?

6.    Starting with the Ohm's law equation and the differential form of Gauss' Law



show that the n+ and n- small ion concentrations vary with altitude as follows:

B+ and B- are the positive and negative small ion mobilities, and e is the charge on an electron.  You can assume the J and E are functions of z only.  Note also that while we mostly assumed that n+ and n- were equal during much of the semester, that is not the case here.

7.    How much charge will appear on the flat plate electric field antenna shown below when exposed to a negative (downward pointing) electric field of 200 V/m?  What voltage will appear across the 100 pF capacitor shown in the circuit below?