But positive z direction is out of the parabola. We used ∂v/∂t × ∂v/∂s, so t cross s is x cross y which is in the positive z direction. As we see, t is equivalent to x, and s to y. We need a systematic procedure to find which way the vector really points (and maybe the expression we get using this procedure contains a and b, so we can generalize without having to get an absolute answer).Īs for this problem, I did find a way around plugging in values, with intuition. In this case, as a and b change, the directions of everything change, so a plug-in at one value doesn't work for a plug-in for another. So maybe v(t,s) =, and we want to see how flux changes as we vary a and b. Sometimes we might not know what the surface looks like so we cannot just observe where the vector points and say, "Oh, that's outward." Also, we might be solving abstractly. Is there some better way to find out where the normal vector points? Other than by plugging in some value. It might be an electric field, and then perhaps there is no net charge inside the parabola. It might be heat transfer, and then there is no net heat entering from the sides (we still don't know about the bottom). If it also equals zero, then yes, the mass of fluid inside is constant.īut if the vector field doesn't represent a fluid's velocity, it might mean something else. To make sure, you would need to compute the flux of fluid through that region of plane. There might be net fluid escaping or entering through that plane. If you just seal the "hole" with a flat plane, then, no, you cannot be sure yet that mass inside is constant. There is a "hole" on the bottom of the parabolic surface. I mean, what region? The surface is not closed. However, we cannot assume that the mass of fluid inside the region is unchanging yet. If the vector field is really velocity of fluid (I think it is in this problem), then the mass of fluid going into the "region" is exactly equal to the mass of fluid going out. ( 4.13) or ( 4.Flux out from the surface is zero. To use Coulomb’s law with such a description, we replace the sums of Eqs. Should be able to do that case before you try to handle the other It will treat only the situation where we canĪssume that the positions of all the charges are known. So although this chapter is toīe on electrostatics, it will not cover the more beautiful and subtle AndĪll of the charges must be taken into account. Other parts from charges that have moved around in the conductor. Know about, from the charge that we brought up but there will be The chargeĭensity $\rho$ in Eq. ( 4.5) may have one part that we If, for instance,Ī charged body is brought near a conductor or insulator, the electronsĪnd protons in the conductor or insulator will move around. The positions that the charges take upĭepend on the $\FLPE$ field, which in turn depends on the positions of Know only that they have distributed themselves in ways that depend on However, we do not know, initially, where the charges are. If we had only to studyĮlectrostatics at this level (as we shall do in the next twoĬhapters), life would be very simple-in fact, almost We will begin with the simplest situations-ones in which the Mike The Feynman Lectures on Physics New Millennium Edition Your time and consideration are greatly appreciated. So, if you can, after enabling javascript, clearing the cache and disabling extensions, please open your browser's javascript console, load the page above, and if this generates any messages (particularly errors or warnings) on the console, then please make a copy (text or screenshot) of those messages and send them with the above-listed information to the email address given below.īy sending us information you will be helping not only yourself, but others who may be having similar problems accessing the online edition of The Feynman Lectures on Physics. This type of problem is rare, and there's a good chance it can be fixed if we have some clues about the cause. which operating system you are using (including version #).which browser you are using (including version #).If it does not open, or only shows you this message again, then please let us know: So, please try the following: make sure javascript is enabled, clear your browser cache (at least of files from ), turn off your browser extensions, and open this page: If you use an ad blocker it may be preventing our pages from downloading necessary resources. If you have have visited this website previously it's possible you may have a mixture of incompatible files (.js. In order to read the online edition of The Feynman Lectures on Physics, javascript must be supported by your browser and enabled. There are several reasons you might be seeing this page.
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