PH214_2009SPRING_HWAL__[0]

PH214_2009SPRING_HWAL__[0] - Essay 1 Question Please answer...

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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 non-zero 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 non-zero 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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loop 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 non-zero 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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PH214_2009SPRING_HWAL__[0] - Essay 1 Question Please answer...

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