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Essay 1
Question
Please answer the following question(s)
in your own words
and
without the use of
mathematical equations
. Limit yourself to one page (being verbose is not a virtue in this
case). Your submission will be graded for factual correctness, logic and clarity of
thought, and proper spelling and grammar. Proofread before you submit.
(Only accepting
submissions as .txt, .rtf or .pdf files)
In class, we developed Gauss' theorem and Stokes' theorem as a prelude to
transforming Maxwell's equations from integral to differential form. We saw (in
Stokes' thm) that the curl has something to do with the line integral around a
closed path. This may suggest to you that the curl somehow has to do with things
rotating, swirling or curling around. This intuition would lead you to (correctly)
conclude that the vector fields in panels A & B in the figure below have nonzero
and zero curl, respectively. However, this intuition would fail you in the case of
the field represented in panel C.
Explain why this uniformly directed field has
regions of nonzero curl. What key piece of information is missing from the
naive intuition described above?
[N.B. for completeness, the equations of the
vector fields are given but you do not need to compute anything to answer this
question]
My answer (original grade 9 out of 10, this was my resubmission after his
corrections)
One of the methods discussed for determining if a vector field contains a curl is to sketch
a rectangular path in a random region and to find the curl around that path.
The path can
be followed either clockwise or counterclockwise.
The vector fields in panel A, along
with any other vector fields that appear to “curl” around a center point, clearly have a curl
when using this method.
The vector fields in panel B, however, do not have a curl.
This
is not because the vector fields do not “curl”, but because the length of the vector
increases or decreases with ‘y’ in the ‘y’ direction (vertical components of the rectangular
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View Full Documentloop have canceling curls).
In panel C, where the increase or decrease in the ‘y’ direction
is dependent on ‘x’, the vertical components of the curl only cancel out when a
rectangular loop is centered at any point where ‘x’ is equal to zero.
This means that a
loop centered elsewhere has a nonzero curl.
The flaw in the naïve intuition is the meaning of a curl.
Intuitively, a vector field “curls”
only if a particle placed at any point in space appears to move in a clockwise or
counterclockwise path, as can be seen in panel A.
While this is one example of a curl, a
curl can also be observed using multiple particles.
For example, place particles along the
‘x’ axes in the vector fields in panels B and C.
After a period of time observe the
location of the particles.
In panel B, the particles form a horizontal line, which is sensible
because no curl is present.
In panel C, however, the particles form a curve.
From the left
side of the ‘y’ axis until the axis, the particles rotate counterclockwise, and from the right
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 Spring '08
 Faculty

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