Gauss' Law Explained and Parallel Plate Capacitor Worked Examples | Doc Physics

so the word flux comes from the Latin Fluxus meaning flow and we're going to talk about the idea of electric flux it does not mean electrons moving from one place to another but it represents the idea of electric field going through an area so it's it's a flowing of electric field more than it is a flowing of electric charge and I've got this machine that I've made it's a few years old so it's kind of dirty but I call it my flux catcher and it is a loop made of metal here you can see that it is a fairly continuous loop and the idea is that if I point it in the right direction it can catch flux going through it I'd like you to think of it as if in this sense when we get to magnetism it will be a little bit different but in this sense I want you to think of it as a screen that I could wrap around I could have it flat or I could wrap it around and form a closed sphere out of this screen flux catcher so if you have a faucet let's put a faucet here we'll put a handle on it and say that out of this faucet is coming some water and if water is coming out of the faucet then you could put your flux catcher right here in this stream of water and if you oriented it so it was like flat like this this by this I mean it's like this direction like like this then you would catch a lot of flux coming out in fact you could even wrap it around the faucet and say that there was flux positive flux coming out of the flux catcher you could wrap it on to make a solid surface let's let's see if I can imagine this can you imagine wrapping a mesh bag around the faucet and find that there would be net flux leaving from that faucet okay and you could also say there's a drain at the bottom of this thing right there's a sink here and help color for sink so if the sink is like that and there's a little hole in the sink here through which water is exiting you could put a flux catcher around this hole make a dome of the flux catcher and you would find that there's a net exit of water through or into I guess kind of into this flux catcher right here so it's about something going through a surface and Gauss oh man Newton would heartily endorse cows to teach you about these mathematical concepts Gauss was a fantastic mathematician and physicist and well let's set up some electric fields you could make electric fields that point in a direction like you could make a steady electric field like this how would you want to make this you guys have an idea to make an electric field that points just to the right that's a good idea because if I have just one charge I get a diverging electric field but this electric field is uniform somehow so I probably have a bunch of electric charges and I'm going to have to say well let's figure out which side is which one of these sides is going to have to be positive and the other ones negative right which ones positive how did I get the electric field pointing to my right would be I could be on my left be positive my left is definitely positive and we're using red to represent that so I get a sheet of metal and I make it positive and this shoot'em if I want the electric fields to terminate here I'm going to have to have a bunch of negative charges over on that side so then I've made an electric field that points uniformly to the right if I put the flux catcher down like that I'm going to catch any flux no in fact but yeah yeah what I find is that the electric field is not going through the flux catcher at all but if I rotate my flux catcher like this then I get maximum flux going through it so there's some kind of an angle that represents yeah it's a sine or coastal it's be careful with it though because you said this was parallel but I don't want to call this parallel there's something that I want to point out about my flux catcher this is the area vector of my flux catcher and it points normal to the surface of the flux catcher so the air of the suckers out and I'm going to argue trying to use this consistently throughout the year that this is the direction in which the flux catcher is parallel to the electric field so the flux catcher catches maximum flux here and the flux catcher catches no flux right here so flux is going to be represented by the capital Greek letter Phi and we'll say that it is electric field times area but we have to figure out whether we want a dot product or a cross product are we thinking there's a sign here or cosine we do want a cosine because we want the area which is to the right right now to be parallel to the electric field which is to the right right now and so we have a dot product this is electric field times area times cosine of the angle between electric field an area where area has to be defined as out of a surface so if I have again we were talking about closed surfaces things that loop in on themselves like spherical flux catchers and in that circumstance the area direction for these suckers is always out that is a strange concept but it's one we've been working with for a little bit and I hope you guys are okay with that that area points out for a bag like that prisons all right so let's use this concept right here and let's also establish now this is this is Gauss's law Gauss's law is a fantastic statement this is electric flux just a definition it is electric field dotted into area and Gauss's law Carl Friedrich Gauss says electric field dot area is flux sure but guess what guys it's also well it's the charge that you've got inside the flux catcher divided by a constant of the universe oh that's pretty simple let's call this the charge enclosed and this is epsilon not it's the permittivity of free space boy that's a long one you can call it epsilon not if you want and epsilon not has this cool property epsilon not is defined to be 1 over 4 pi times Coulomb's constant which makes a lot of sense and electricity right yes it does yes so this is just another way of defining this and I want to do some examples of Gauss's law now this is how our book presents it in the book doesn't even have the dot product but you know that we want to study a little bit calculus on the way so we're going to say that flux is actually the integral of electric field but we have to dot it into a differential area and we have to have an enclosed surface we have to be going over some area the surface area of our Gaussian surface so we create a Gaussian surface in our minds saying this is not something we're making but we're going to say in our minds we can find well Gauss's law is incredibly powerful because we're going to be able to find the electric field from Gauss's law we'll be able to measure the charge that's enclosed and we can find the electric field of anything based on this really really powerful mathematical tool so let us get ourselves a charge and here it is we'll call it Q 1 and we're going to make a Gaussian surface and Gaussian surfaces are always made so that their symmetry makes the integral fantastically simple in this case do you know which way the electric field is pointing around this charge well less is a negative charge and just to spite you that's what I'm going to make it it is in everywhere very good or very wrong so the electric field is always pointing in here and that that implies that we should make a certain shape of Gaussian surface what do you suggest a sphere yeah it's actually got to be a three dimensional surface so we're going to make a spherical Gaussian surface now remember these lines go out to infinity or rather come in from infinity dang I wish you'd said the other directions okay so there's some radius of our Gaussian surface and that actually doesn't matter we're going to get the same answer for the electric field regardless wait a second maybe that's not true doesn't the radius the electric field tell us that what do you expect you expect a strong electric field near the charge or a strong electric field far from the charge yeah so we're going to find that the electric field at this location of the sphere of the Gaussian surface sphere is going to depend on the distance that the sphere is away from the center charge so our our integral looks like this it says let us integrate around a sphere of the electric field dotted into the area and we have to do it over the entire surface area of a sphere oh boy that's tricky now the way the reason we chose the symmetry is the electric field here is the same as the electric field here and there and there and there and there and there right that makes this fantastic what can we do to the electric field variable if it is constant throughout our integration we can pull it out of that integral so we get the electric field at each of these locations times well oh we need a simple a simplification because we said electric field is in the same direction as the area in this case we get a minus sign right because the electric field is in but the area is out that's a little bit annoying and we'll just be sloppy about it will just say that electric field times the integral over some sphere of the area of the sphere well that simply tells you to find the surface area of a sphere and you've been taught that right surface of a sphere is what mm-hmm now we know Gauss's law says let's write down what Gauss's law says it says take the enclosed charge and divided by epsilon naught so I'm just going to write the Vettes enclosed charge divided by epsilon naught and then I'm going to say well II times the area of a sphere is look at this I have this written three different times and it's exactly this but on the next line I'm going to say the surface of a sphere is 4 times pi what else there's an R square there and this is just the charge that's enclosed now in this case we can substitute in what we know to be actually the enclosed charge it's q1 and it's negative and then we have to divide by epsilon naught and if you want to know the electric field at some distance R around a point charge well it's just the charge that's inside divided by 4 PI epsilon naught R square that's the game that's it right there but you remember that 1 over 4 pi KC is epsilon naught so you can change the location of those two guys and this is just KC times Q divided by R square you ever seen that before I think that you have you have seen the electric field in point charge and that's what it is notice this epsilon naught in KC thing and the 4pi is jump around all over the place cool now I want to do Gauss's law in one more example and that's the example of these two parallel plates I want to get you a well this could be this could be a model of a capacitor force so let's draw ourselves another capacitor and we will set up Gauss's law and there's another symmetry choice that we make so that the electric field will cancel out in some locations and be awesome in other locations you guys ready for me if this is the last example if you okay everybody all right uh-huh so here we've got electric field point in the same direction as we had previously so we need to have some negative charges over here to terminate the electric field and we need some positive charges over here to begin the electric field and I need to define something called surface charge density I'm going to use ro to define that I'm going to call it surface charge density because as you guys will see on the first video the hopefully you have seen it when you're watching this the surface charge density is the only charge density for a conductor because that's where the charge is lie they lie on the surface of a metal surface charge density is Sigma and so my plan is to take a Gaussian surface that looks like this I want a cylinder and I want it to go around this plate now remember Gaussian surfaces aren't real they're a construct of the mind so I'm going to give you an area a right here and an area a right there and I'm going to argue that it doesn't actually matter how big the circle is do you see why let's consider what this flux is flux is electric field dotted into area and then add it up completely all of the electric fields added into the area which way is the electric field ooh which mozi lecture field over here no electric field at all because the electric fields entirely within the two plates of a parallel capacitor uh-huh and what about which wave electric field over here boom so as any of the electric field leaving the sides of the cylinder no none of there so there is no flux through the sides of the cylinder so we could say flux separate this out in two parts we could call this part one and we could call the sides part two and we could call the top part three so let's see we've got the sides and those are flux two and that zero because the electric field is normal to the area in that case the area points gets area vectors here the area is pointing directly out and the electric field is pointing that direction so it's always going to be normal so you won't have an electric field there you won't have a sorry we'll have flux you've got an electric field and the reason that the flux is zero over here is because the electric field is in fact zero outside of the parallel plate capacitor but surface three does have a flux there is electric field going through there and we know that the total flux for this Gaussian surface well we know that the total flux for the Gaussian surface which is all five three because those guys are zero is going to be when I said it's going to be charged and closed divided by epsilon naught we also know that it has to be the integral over the area of electric field dotted into area and the wonderful thing is the electric field sorry we need a differential area right there that's da we know the glacier field is in the same direction as the area's pointing here so we can simplify that to just e D a because they're the same direction won't have any vector problems anymore and we're trying to find the electric field but guess what do you think the electric field is the same everywhere on that area yes in fact it would be true the electric field is everywhere the same inside of here so yes the electric field throughout that area is the same and what can we do when there's a constant inside of our integral pull that sucker out so I'm going to say that e times the area of a circle ah-ha-ha-ha is equal to the enclosed charge divided by epsilon ma I want to expand this right side though the enclosed charge has to be the surface charge density times ooh the enclosed charge has to be the density times what yeah the density times the area of the circle and then I'm going to divide by epsilon naught and what about this left side over here I took out a already and this says integrate the area differential element over the area which gives me when I guess the area so this is electric field times area and this is Sigma times area over epsilon naught so it doesn't matter size it doesn't matter the size and we find that the electric field inside of a parallel plate capacitor is simply Sigma divided by epsilon on mmm that beautiful yeah physics put that here for you flowerpot

Channel: Doc Schuster

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Id: hnt07JpNE1E

Length: 17min 27sec (1047 seconds)

Published: Mon Jan 07 2013