Physics 230 February 13 2002

Exercise on analyzing the result of a calculation.




In class we obtained the electric field produced by a sheet of charge lying in the $x-y$ plane, with charge density $\sigma $ that is infinitely long in the $y-$direction, and extends from $x=-a$ to $x=+a.$ On the $z-$axis the electric field is given by:
MATH
Now we want to analyze this result.




1. Dimensions. The angle is just a number with no units$.$ The charge density has dimensions of charge/length$^{2},$ and so the electric field is MATHcharge/length$^{2},$ and these are the correct dimensions.

2. Check the result a large distance from the strip.

At a great distance $z\gg a$ from the strip, the strip looks like a line charge with line charge density MATH Thus we should find (Example 24.2) MATH In our expression, $a/z\ll 1,$ and so MATH and we have
MATH
as expected.

3. Check the result very close to the strip. For $z\ll a,$ the strip "looks" infinitely wide, and so we should obtain the result of Example 24.5, MATH In our expression, with $a/z$ very large, $\tan ^{-1}a/z$ is almost $\pi /2,$ and so
MATH
as expected.




The result passes all three checks and so is probably correct.

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