![]() ![]() However, if I apply this boundary operator again, I get zero, because the boundary of a sphere is zero, or another way to see it is the boundary of a boundary is zero ($\partial^2 = 0$). Yet again, if I took a ball this time, its boundary would be a sphere, and so a ball is not closed. On the other hand, if I took a sphere $S^2$, then its boundary is zero, and it can completely enclose a portion of charge. We say that $\Sigma$ is closed if and only if $\partial \Sigma = 0$ which is to say it has no boundary.įor example, if we took a two-dimensional disc of unit radius, then its boundary would be a circle, and so $\partial \Sigma = S^1 \neq 0$ and so it would not be a closed surface. Let $\Sigma$ be a smooth sub-manifold in $\mathbb R^n$, i.e. ( 4.13) or ( 4.Forget outside and inside. 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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