Gauss's Law 02 |
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OPENING QUESTION: Write down the various permutations of Gauss's Law on one sheet of paper on a single sheet of paper (we'll use this to write, rewrite and otherwise edit our own personal working model for all things GAUSS) On another blank sheet a paper write down a list of questions that come to mind about when and how to apply Gauss's Law (write them in full sentences please, not in shorthand) As we go through various permutations of Gaussian Surfaces and such, you'll need to edit/rewrite your model (page 1) answer questions you've posed (page 2) and add new questions that arise (page 2 again) Remember, this is brand new stuff... being baffled is ok, we'll make good progress today (promise!) OBJECTIVE: I will continue working on formulating a model for using Gauss's Law during today's class. WORDS/FORMULAE FOR TODAY Terms:
Constants:
Formulae:
WORK O' THE DAY: Now let's add yet ONE MORE item to your overflowing plate of Gauss! There is an important geometric difference between these two images... can you find it?
═══════════════════════════ So... let's recap our permutations of Gauss's Law using our more accurate integral notation: ALWAYS start with a basic recollection of Gauss's Law: The electric flux through an enclosed space is equal to the sum of the enclosed charge divided by permativity of free space Which we express mathematically as: Φ_{E} =q/ε_{o} By itself, that is pretty boring stuff. However, recollect that we have another more power equation: Φ_{E} =∮EcosθdA Setting those two formulae equal gives us: q/ε_{o} =∮EcosθdA ═══════════════════════════ Let' start with that equation as our *Standard Model*. You'll need to and apt/develop your own model further from now on. For now, let's take a fairly easy example and and apply our model to that situation. Consider a cube with electric field lines moving parallel to the x-axis in the positive x direction as shown below: How do we calculate the electric flux through that cube? 1) Please start by adding rules to your page 1 that you think you should apply. Any questions that arise should go on page 2. 2) Without consulting your group (yet), suggest an answer to the problem and provide backup to your response. 3) Compare your work with your group and make additions/deletions to your model. Cross out (but do not erase) questions that you answered and add new questions that arise. ═══════════════════════════ We can answer this qualitatively as follows: Gauss's Law applies to charges enclosed by a Gaussian Surface. The cube does not enclose any charges therefore it has zero electric flux. We can answer the question with a bit of math as follows: The efield lines enter the Gaussian Surface (the cube) and an angle of 180^{o} to the area vector of the left side of the cube and exit 0^{o} to the area vector on the right side of the cube. cos0^{o} and cos180^{o} will cancel giving us zero electric flux. There are no electric field lines moving through the top/bottom (yz plane) or the other two sides (xz plane) so there is zero electric flux there. Therefore there is zero electric flux. Please adjust your model and questions once again. Add new questions, cross out old ones and make changes to your model as appropriate. ═══════════════════════════ Take a look at the following images and note any specific observations that you think will be helpful in evaluating the electric FIELD on the surfaces S_{1}, S_{2} and S_{3} of image #1, and the surface of image #2
Ok, so I know what yer thinkin': "We just derived the equation for finding an E field... how is that helpful?" Indeed...! ═══════════════════════════ Let's revisit our favorite "Long Thin Wire" scenario. Let's say we have such a beastie with uniform charge density *yay*.
Something like this, mayhap?
Please make any appropriate adjustments to your page 1 and page 2 as appropriate. ═══════════════════════════ Now let's try some reverse logic (ie, eg, for exampe, whatever) a scenario that looks promising but turns out to be inaccurate and misleading... work with your group to see if you can determine the bogartry of the situation: Let's say we have a +5.0 μC charge and a -13.25 μC charge and a +7.2 μC charge in close proximity and we want to find the value of the Efield at 1.0 m distant. Sketch the charges thusly: Now choose a Gaussian Surface surrounding those charges. Now we add up all the charges and solve the problem yay? Nay! Hint: Use our animation tool to show the efield lines for such a situation. Does that help? Let's try some fill-in-the-blank: We can use Gauss's Law to find the ________________ in the above situation but we CANNOT use it to find the ______________. That's because Gauss's Law allows us to sum up the charges when we find the _____________ but it also requires a high degree of ________________ when we use that law to find _____________; and in this case that high degree of symetry is clearly lacking which means the Efield is NOT _______________ which means we've got a *nasty* integral to deal with and we have to punt! ═══════════════════════════ |
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Please work on the following problems with your crew: Objective problems 8, 9 & 11. Conceptual problem # 7. Problems 7 & 8 |