Chronology |
Current Month |
Current Thread |
Current Date |

[Year List] [Month List (current year)] | [Date Index] [Thread Index] | [Thread Prev] [Thread Next] | [Date Prev] [Date Next] |

*From*: John Denker <jsd@av8n.com>*Date*: Tue, 02 Apr 2013 19:23:59 -0700

My father's favorite joke:

Q: What's green and has six wheels?

A: Grass. I lied about the wheels.

He had about 100 more jokes in the same category. He thought they were

hysterically funny.

Also:

Q: How do you calculate the capacitance of a cow?

A: It's easy. Start by assuming a spherical cow.......

I mention this because on 04/02/2013 04:24 PM, Carl Mungan wrote:

I am planning to put the following problem on a test tomorrow and I'm

somewhat surprised by the answer. So let me ask the list for insight:

A square waveguide extends from -infinity to infinity in the z direction.

It has two grounded plates at y=-0.5 and at y=0.5. The plate at x=-0.5 has

potential -sin(pi*y). The plate at x=0.5 has potential +sin(pi*y). What is

the surface charge density on the plate at y=-0.5?

(Actually the first part of the problem asks one to find the potential

everywhere inside the waveguide. I give them the hint to just write down

the appropriate form to match the boundary conditions. For those a bit

rusty on this kind of problem, this means a sinh in the x direction and a

sine in the y direction, with each coordinate multiplied by pi. You can now

find the overall scaling constant by fitting to the boundaries.)

This strikes me as seriously ill-specified. It's something of an ESP exam,

because it hinges on reading the instructor's mind, to find out what was

intended.

There may be a cultural issue here. It is important to know the rules. As

the saying goes, if you're supposed to be playing football, don't show up

with your baseball bat and glove.

I assume today's game is supposed to be played by ivory tower rules. That

is, we are supposed to fill in all the missing information by making the

simplest and most favorable assumptions.

Tangential remark: I hope that at some point, before these students go

out and take command of nuclear submarines, they learn to play by different

rules.

Returning now to the spherical cow in the ivory tower:

1) I assume the word "waveguide" means one thing and the word

"plate" means something else.

2) I assume the boundary conditions are independent of z.

3) I assume the desired solution is independent of z.

4) I assume the size of the waveguide is 1 unit in whatever units

we are using.

5) As a consequence of (2) and (4), I assume the "plates" are

coextensive with the walls of the waveguide. This contradicts (1).

6) I assume the boundary conditions are independent of t. This is

sort of implied by the word "Laplace" in the subject line.

7) I assume all the plates are perfectly conducting.

8) I assume the boundary voltage is as stated.

Now we have a problem, because the specified voltage pattern is inconsistent

with the assumptions. The voltage distribution is about what you would expect

for a traveling wave in a waveguide ... but the Subject: line suggests DC. So

where's the gag? Is it not Laplace? Is it not DC? Is it not a waveguide? Is

it not green? Does it have less than 6 wheels?

I'm not trying to be nitpicky here. The first time I read the question, I

thought the waveguide was much larger than the plates, i.e. the plates were

just little antennas sticking into the waveguide to launch the wave. I assumed

that the switch from "waveguide" terminology to "plate" terminology must mean

something. It was only after seeing the intended solution that I was able to

work backward and figure out what the question must have been.

I predict that students are going to waste huge amounts of time trying to

decode the statement of the problem. Everybody likes "rich context" problems

when the context is real enough that you can use real-world intuition. OTOH

when real-world terminology such as "waveguide" is painted onto a completely

fake problem, the more you know about the real world the less able you are to

solve the problem.

My suggestions:

-- If the problem is z-invariant, say so.

-- If the problem is t-invariant, say so.

-- If the boundary conditions are incompatible with real-world waveguides,

don't call it a waveguide.

-- If the boundary charge distribution is arbitrary and artificial, say so.

-- If the plates are coextensive with the walls, pick one set of terminology

and stick with it: either 100% plates or 100% walls. Failing that, at

least define the terms clearly enough that we can see that they are

interchangeable.

-- If *one* pair of walls is perfectly conducting and the other is not,

say so.

=====================

If the original question meant to ask for intuition about the sinh .....

1) My response to that is, what else could it be? By symmetry it has to be an

odd function. How many odd functions are there that are the solution to a

homogeneous second-order linear differential equation? The first things I

would guess are

a) straight line

b) sin

c) sinh

2) In more scientific terms:

-- This is a 2D problem. The z dimension is just there to distract the tourists.

-- The 2D Laplacian says that the curvature in the x-direction has to just

exactly cancel the curvature in the y-direction. Every point is a saddle

point. This rules out (1a) and (1b) above.

3) Perhaps even more fundamentally: The crucial "break" in the process of

solution is guessing that there might be a separation of variables such that

the solution can be written as a product of an x-function and a y-function.

There is no law that says such a factorization is possible in general. By

way of counterexample, the atomic p-orbital is certainly not factorable

into an x-function and a y-function. To say the same thing the other way:

the pseudo-waveguide situation was artfully contrived to *make* factorization

possible.

So in some sense, the main thing the question is testing is simply this:

a) Can the student decode the question?

b) Will the student be clever enough to try a product-form Ansatz? If yes,

the rest of the work is trivial. If not, things are going to get ugly.

**References**:**[Phys-L] Laplace equation problem***From:*Carl Mungan <mungan@usna.edu>

- Prev by Date:
**Re: [Phys-L] Laplace equation problem** - Next by Date:
**[Phys-L] Counterproductive Targets** - Previous by thread:
**Re: [Phys-L] Laplace equation problem** - Next by thread:
**Re: [Phys-L] Laplace equation problem** - Index(es):