Lecture 07 Spherical Charge Distributions

welcome back to physics 272 I feel that way okay alright so last time we calculated the electric field of a ring and of a disc and we we did this in the usual way right we pretended that the ring was a bunch of point charges and then for every little point charge we calculated the electric field that our observation point by symmetry we throw away the components that we're gonna sum to zero so we didn't have to calculate them then we did some calculus thank you mr. Newton we did some calculus and then we found out these equations so we've got the electric field of a ring along its axis now I haven't put the direction here but you can figure out the direction right if it's a positively charged ring then the electric field should point away if it's a negatively charged ring then the electric field should point toward the ring and we did the same thing for the disc for the disc you can either think about adding up the concentric rings which is one way to do the calculation I found it a little easier to start from scratch and go from point charges but you can do whichever makes more sense to you and in this case we got a more complicated looking equation but still pretty similar that it points away from the disk if it's a positive charge and it points towards the disk if it's a negative charge then having built these guys up we took the size of the disk to infinity so we let the radius of the disk go infinite but we did it in a particular way right we let the radius go to infinity we let the charge go to infinity but the charge per unit area we held constant and then we got that for the infinite plane the electric field coming off of an infinite plane is uniform okay so it's uniform and then it's got the same magnitude everywhere in space and it always points away from the plane if it's a positive charge and towards the plane if it's a negative charge where Sigma here is charge per unit area so anywhere I look in this plane if I take a little chunk out of it and measure the charge on that piece and then divide by the area of that piece I get the same number charged per unit area and this was a bit weird that the electric field comes off and just goes with no abatement right so as far away's you'd like to be from that infinite plane you still get the same electric field that's weird we hadn't encountered that yet where there was an object that we could get arbitrarily far and still feel the same electric field so that's weird but it's because there's an infinity involved as an infinitely large electric sorry there's an infinitely large sheet charge an electric have a plane of charge right there so one of the other things to think about is how the electric field how the arrows point in space so for a point charge we had the electric field coming off the point charge and the arrows kind of come out like a dandelion or a starburst shape and then as the arrows get farther apart from each other the electric field gets weaker and weaker okay for this plane of charge as the electric field is coming away from the plane of charge the electric field is just coming straight away okay and the only direction you can go away from an infinite plane is straight away right it doesn't splay like the point charges electric field does so it just goes straight away and because it's going straight away there's always the same density of field lines any distance away so that's intuitively why this electric field doesn't decrease as you get away from the plane it's a weird result okay but it's a true result do you have any questions about how that stuff went okay and then we saw that if we add up to infinite planes next to each other one with positive charge and one with negative charge then in between the electric fields add because the arrows are pointing in the same direction it points away from the positively charged plane and towards the negatively charged plane and so that was a reinforcement effect we got twice the field inside and then outside the fields canceled completely so that there's zero electric field outside of two parallel oppositely charged infinite planes now this is actually a bit like a capacitor a capacitor has two two plates that are oppositely charged and if the plates are large compared to the distance between the plates then inside you can pretend you have infinite plates and so inside the electric field is about Sigma over epsilon and outside the electric field is very small in a real capacitor there'll be a little bit of electric field leaking out we call that the fringe field but it's pretty small okay do you have any questions about that location okay all right now I want to give you some guidance as to how to approximate we've done some situations where we make approximation such as the infinite plane or getting close to a disc or close to a rod or far from a rod so I want to give you some guidance as to how you know when a particular approximate formula is a good thing to use when should you use it when should you not use it and and in all these cases we have two length scales in the problem when you have two length scales in the problem then you can compare the two and if one is significantly larger than the other then it's a good time to think about making an approximation okay so for example if I have a point charge and I'm a particular distance from a point charge the only length scale the only distance in the problem is how far am i from the point charge because the point charge is zero extent it doesn't have a length of its own now if I have the case of a rod or a disc or a sphere something like that now I'll have two length scales in the problem I'll have how far away am i from the object that's one length scale and then I'll have how big is that object that's the other length scale so anytime there are two length scales think about whether you can make an approximation and the way to make it just be crystal clear is to pretend the linked scales are money okay just convert into new units where it's money and then think about having one linked scale represent billions of dollars and the other one represents pennies right if you have two friends and one owes you a billion dollars and the other owes you pennies who do you go after right you go off for the pretty well you go off for the person who owes you a billion dollars and then once you have your billion dollars do you even care to go collect two pennies from your friend who owes you two hopefully you don't you know come on I don't care what do I care I'm a billionaire here have some extra pennies right you don't care okay so if you see this situation where you going to add together something that's like billions of dollars and add pennies to that don't even bother with the pennies right if you were a billionaire I guarantee in your checkbook when you do the that I don't even know if you do that anymore of things online but anyway when you keep register of things you wouldn't even bother keeping track of pennies so when to approximate let's say for example we have a rod so let's have that we've already calculated the electric field of a finite rod and the rod has a links to it particularly and then I'm going to be a particularly scales on the problem the size of the rod and how far away am i from the rod so would have two length scales I should think can i approximate here so let's say that we have a rod of a hundred meters in length and I'm going to be a distance of one millimeter away already my diagram is not the scale right if I drawn it to scale you wouldn't even see the separation between the rod and the point okay so always do that always draw a diagram and then think about what what is the ratio of these lengths here now in this case the distance from the rod x over the length of the rod those those two linked scales the ratio of them is very small so ten to the minus five that's much less than 0.1% that's my cutoff for these things if the ratio of your lengths is 0.1% or less think about doing an approximation it's a good good time to do that so here's the full equation the exact expression for the electric field coming off of a rod if you're in the bisecting plane of the rod and I can see that I have an expression in sight you know inside of this I have an x squared plus an L over two squared that's what you're looking for you're looking for the plus sign so find the plus sign somewhere if there's two different length scales on each side of the plus sign and one is wildly larger think about making an approximation and here's how you do it pretend they're money okay tend their money ones billions of dollars in one's pennies right and this guy screams show me the money in my example where's the money where's the billions it's on the L the billions is on the L so if the billions is on the L that's where the money is don't care right this guy's like billions of dollars this guy's like pennies who cares about pennies what's a few pennies among us billionaires right hmm we don't care that one that'd be nice so you just X it off right who cares about the pennies because I'm adding pennies to billions right so exit off now can finish your math having taken care of that term so these were added together I crossed off the really small term and then I write down the math neglecting that really small and then I continued on okay that's how you make these approximations you need to find two terms added together where one is like billions of dollars and the others like pennies ya X off the pennies question Oh excellent questions so he wants to know why when I went back here why did I focus in on this X why don't I when I cross F X everywhere so I can only cross things off when I have two things added together so the two things added together a billion plus pennies is still about a billion but some number times pennies is gonna reduce the whole thing so that's a good question I do look in the expression for the plus sign so there's the plus sign it could be a minus sign sometimes but look for the addition term and look on either side of that if one's really small just neglect it okay but you're right I can't just neglect the X everywhere yeah good question all right other thoughts about this I hope this gives you some guidance as to you know we're kind of trying to train you in the homework as to when you can make a good approximation and when you shouldn't make an approximation so if you have a situation with two length scales and one is wildly larger than the other look at your expression your exact expression and see if you can make an approximation and this is this is how you do it you find the plus or minus sign you neglect the side that's pennies cross it off carry the math through and then that gives you the approximate expression and it also tells you when it's okay to make that approximation okay any other questions or comments about that okay all right so new material for today we're going to look at spheres we're all about spheres today we've done a rod a disc a ring we're going to do spheres now and we'll see that we'll have a hollow sphere of charge and we'll see what the electric field is for that and then we'll have a solid sphere of charge and we'll see what the electric field is for that in the hollow sphere case we're going to assume that I simply have a spherical shell and there's charged uniformly distributed all over the surface of the shell in the solid sphere case I'll assume I have the same charge but I'm going to uniformly distribute it all through the interior of the the sphere so in both cases it's helpful to think of these things as being insulators so the general procedure that we've been using is to cut that charge distribution up into little point charges and then we calculate the electric field did one point charge then we sum up all the point charges and as we do that we sum up all the electric fields take an integral and then we get an answer okay and that's of course a valid way to do this and I can show you all that math but you've seen it a few times plus in this case it's actually really hard so it's more work than we want to show you in lecture so we're going to take the other method which is since it's a lot more work we're going to use symmetry skip the actual calculation although it does work out and it gets to be a bit like this cartoon here you know I think you should be a little more explicit here in step two there's the magical step than a miracle occurs so for now you'll just have to say look if we actually did all the math that would formally come to zero all right later on in the class we'll learn a really nice trick called Gauss's law that will let us prove it for real and a much quicker way so we'll do that that all right so let's think about then using symmetry we'll go back to symmetry principles and think about what is the symmetry of a sphere so one of the things we need to think about is the electric field should have some sort of shape to it that is coming from the shape of the sphere itself so watch this I'm gonna rotate here's a sphere okay I'm gonna rotate it all right are you ready go did you see it okay I rotated a sphere and it looked just like a sphere right and then I rotated the sphere and it looked just like a sphere here's the nice thing about when we take the physics and turn it into math okay when you translate it into math if there's no difference in your diagram if there's no if you can't see the difference when you rotate the thing the math can't tell the difference either so in physics if you can't tell the difference there is no difference okay so if I rotate this sphere it's the same physical situation as the original sphere right that means the electric fields gonna have that kind of shape such that when I rotate the electric field it looks the same as if I hadn't rotated it now another thing we need to take into account is that from very far away every charged object looks like a point charge okay so if I take that sphere and move it very far away from us into the next building you know it's still there but from very far from the sphere it looks exactly like a point charge so far enough away we need to recover the point charge equation for the electric field okay and whatever it is up close it needs to also have that symmetry of the sphere such that what I rotated it looks the same so we'll combine these two ideas that the electric field has the same symmetries as the sphere that it simply has to be some sort of radial distribution it can point out of the sphere it can point towards the sphere but it can't point sideways because if it pointed sideways I'd be able to tell if I rotated the sphere right but if it points towards you and I rotate the sphere now there's another port piece pointing towards you that sort of thing so it's got to look like a point charge from far away it has to look the same if I rotate the sphere and so this is the solution that does that basically outside of the spherical shell the electric field takes the same form as the point charge and it just always looks like the point charge even if you're really up close it still looks like the point charge okay and you had a question already did I answer the question no you got it okay all right so the electric field is the same symmetries as this as the sphere and the result is that outside of the sphere the electric field looks as though there were a point charge sitting smack in the middle any any questions or comments or complaints about that okay so here's the question question is a charged sphere with charge Q sits a distance D from a point charge okay assume the sphere is insulating by the way is the force that the point charge Q exerts on the sphere the same as the force that that point charge would exert on another point charge big Q so this little charge is exerting a force on the sphere how does that force compare to if I didn't have the sphere but I had a point charge of big Q sitting at the center okay what are some things we need to consider in order to solve this problem what are you thinking or you can always tell me what your neighbor is thinking and then maybe your neighbor won't talk to you next time that's okay all right so what's either either an answer or a line of reasoning that we should draw out and think about yes please okay so I'll just repeat it for people in the back you chose yes because the electric field of this sphere is the is identical to that of a point charge so in some sense this should be identical to that okay yo please right here oh okay all right so I'll repeat that for people in the back so you said look I didn't give you a link scale I just said it's it's outside at some distance and it's outside so then I can take the limit where they're really far apart and then the limit where they're really far apart it's really clear that the sphere looks just like a point charge so they must be the same yeah it's always good it's always good to think about those those limits actually I think I think that really clarifies things a lot right you come up with an answer and then you think about it an extreme limit and you see if your answer is robust against those extreme limits I think it really clarifies things yeah good other things we should take into account okay what about okay so so so far we've discussed what the electric field is it Q at the little point charge Q right but really we want the force on the sphere don't I need the electric field at the sphere in order to calculate that that's force kinda so I think you used another principle to finish the problem because because so far worth we're thinking in terms of the electric field that the sphere puts out and what it looks like at the point charge Q and that gives me the force that the sphere exerts on the point charge but we want the other direction yeah okay reciprocity so we know somehow whatever this is it's got to be equal and opposite forces so yeah we put that principle in as well so that if I know what the force is on the point charge must be equal and opposite of the force on the sphere yep good okay I think we drew out all the physics is there something else we should think about okay so inside the spherical shell this is the magical step that I'm not proving to you but the answer is that inside of a spherical shell as long as this is an insulating spherical shell and the charge is uniformly distributed and can't move then the electric field that this sphere puts inside of itself is net zero okay so all the little charges contribute something right each charge contributes some electric field inside but if I sum up all the electric field vectors so just thinking for example at this spot be inside the sphere there is some influence at it due to this point charge here number one it's actually pretty big since P is close to one and so there's an electric field pointing to the left there from charge one from charge number four there's an electric field pointing to the right it's a little weaker so those don't quite cancel from point charge two there's an electric field down from point charge six there's an electric field up those cancel and then from this whole side of the sphere there's a bunch of vectors pointing to the right from this whole side of the sphere there's a bunch of vectors pointing to the left it's a complicated summation but in fact it all sums up to zero in the end so inside the electric field of sorry inside the sphere the net electric field due to all those contributions sums up to be zero and it's a long calculation that I'm just not showing you the guts of and instead I'm saying then a miracle occurs all right and later when we use Gauss's law we'll be able to see exactly why that's the case with a much shorter calculation okay so for now what I want you to know is the result that inside the spherical shell the electric field is zero so it means if I were measuring electric field and I were outside the sphere measuring electric field I would detect something if I'm inside the sphere measuring let's say I'm inside and I don't see the shell and I'm trying to detect whether it's there just by measuring electric field I wouldn't know okay it's as though that shell is invisible to me all right so here's the summary for that the electric field of an insulating spherical shell from the outside for an observation point little are larger than the radius of the sphere it looks just like a point charge located at the center if I'm inside if my observation point little R is inside the sphere it looks like zero question okay so on the surface itself let's not consider on the surface itself okay let's have some really sharp boundary to where we can say that we're either outside or inside and we'll let the surface itself be just a boundary okay I see you are a math major or it could have been okay all right okay so we'll just that boundary right there we're gonna either say you're outside or inside that it's so fin you can't sit right on it okay and that's kind of a physicists way to answer that question a mathematician would answer a little bit differently okay right actually it really you can choose it either way you like there's only one point there right as a function of distance there's just one point and so you'll get the right physics answers either way you want to look at it if you want to look at that point being zero you'll get the right answers for everything if you want to look at it as connecting to the outer one it's it's okay yeah because we'll you know from the outside it's all going to look the same or from the inside and all with the same yeah good question all right so inside the insulating spherical shell let's just think through the physics consequences of that okay so the physics consequences of this is let's think of an insulating spherical shell so I'm not gonna let that charge move around now I want to put a charge inside I'm just going to take an extra charge and put it inside okay what's the net electric field inside of the sphere now okay there's a negative charge there so the negative charge is putting out an electric field so there must be something there now so the the negative charge exerts its electric field which adds to the zero contributed by the sphere yeah right here okay all right so he's bringing up the point of well what if we had a conductor if I had a conductor if the spherical shell was a conductor then the charges could move around and I'd have to think about you know that would be a different situation right you're right so we're gonna make this guy for now we're gonna ask your question in like two seconds so for now I want to keep this guy insulating and if the spherical shell is insulating then the charges don't run around on it they just stick wherever I put them and they'll be there yes but keep that in mind because I exactly where I went ahead so for now with an insulating spherical shell there's there is an electric field in there it's due to the point charge what's the force though on that on the negative charge what's the net force on that guy what is it zero okay and why is it zero yeah yeah okay so I can think about it either in terms of okay there's all these forces on it from all the charges out there but they must cancel out because the electric field at that position cancelled out at least due to the sphere so the the the charge the point charge doesn't exert a force on itself we just care about all the other fields and so the fields that are produced external to itself the fields that are produced by the sphere all come to zero field so then there can be zero force on it so that's kind of weird I can put a point charge anywhere inside of an insulating charged sphere and as long as that insulating charge sphere has its charge distribution uniform and distributed there'll be no effect I could just put the charge there and it won't it won't move okay now what if we think about the case where the sphere is conducting this is the the question you brought up in the back okay all right so now we'll make it a little more complicated now I have a conducting sphere I put charge on the conducting sphere what do these charges that can now move around because it's a conducting sphere how do those charges respond to the negative charge I put inside so I've got this little negative charge down here to the right which way do you expect those positive charges to move yeah they're going to want to move towards it a little bit so if I have this guy being a conducting sphere that's where the charges can move if I put a negative charge here the positive charges will move toward it they won't all run to the same point because they repel each other too but there'll be a little net excess positive charge distribution and then once that happens now what's the force on this guy on the point charge pardon well it'll tend to it'll it'll get attracted right so so the positive charge you're gonna Chuck is attracted to the negative charge if positive charge gathers over here that'll give me a net electric field now inside the sphere which will then pull that point charge over so the point charge will be unstable if it were if it's a point charge inside of an insulating sphere it won't feel any forces for the conducting sphere the conducting sphere will rearrange itself and then it'll suck the point charge onto the surface okay questions about that yeah okay so so you're asking the question of let's think of the point charge and what influence it exerts on the shell and what kind of force does it apply so if if the if the negative charge is inside of an insulating shell it actually won't be allowed to exert a force it looks like it should but you can use reciprocity to tell you that well the point charge isn't feeling a force therefore it's not exerting a force on the sphere okay if it's a conducting one it's different right yeah question here good good thoughts oh okay that's a good question so you're taking this one step further you want to take my let me see if I understand you want to take my set up here where it's an insulating sphere of charge put the negative charge in the interior and now what's the field outside is that the question well okay now this is a good question and and so the thing is that in order to find net electric field I always need to think in terms of think in terms of the electric field at the point charge puts out and it's all over space the right fields don't stop they just go all over space okay so the point charge is exerting some you know starburst shaped field all over space the sphere is also exerting outside of itself it exerts a starburst shaped field okay so that outside I'll get the summation of those two so I will get something yeah so the point charge still exerts its own field even outside of the sphere if it's an insulating sphere yeah exactly so if I wanted to yes so he's talking about how to get the addition so if I wanted to get the total field out here for this physical configuration that I drew here I would write down the the electric field due to this point charge and then I would pretend the sphere was a point charge concentrated at its center write down that electric field and add them up directly and I'll get a contribution from both absolutely yep yep this is exactly the kind of toy problems that we like you to go through that me you know helps make sure that you understand all the concepts yeah question here in this configuration the net electric field inside the sphere would be just that due to the point charge so it turns out that whenever you have these spherical shell situations as long as the elect as long as the charge is uniformly distributed all over the shell as it is in the insulating case if you're inside the sphere you may as well neglect the sphere it's as though that sphere doesn't exist you can just poof just disappear have this fear disappear okay yep good okay good question so to clarify this let's think about two spheres okay so what I'd like to build up to eventually is I'd like to eventually build up to a solid sphere but for now we're just gonna start with two so let me have two spherical shells and I have some spherical shell here that's got a charge distributed make them insulating so the charges can't run around and then I've surrounded it by another spherical shell that's also insulating and has some sort of charge in it okay put the same charge on each of them and have them be concentric so their centres line up okay I have two concentric insulating spherical shells they each have the same amount of charge on them cue make it positive okay so they each have a positive charge Q on them I'd like you to tell me what's the net electric field in between the spheres okay so tell me what you're thinking how should we answer this question how should we think about this what do we need to take into account what are some thoughts yeah so you use the fact that this outer sphere is an insulating spherical shell the charge isn't going to move around on it the outer sphere is exerting zero field and since I'm sorry it exerts zero field in its interior since I'm at a point that's interior to that sphere I may as well throw out the outer shell and then answer the question and then it became clear and you said okay then just the inner shell contributes and you got B yeah okay this is good okay so and I like this we should always think in these various limits well alright so what if I had really thin spherical shells and maybe I could go to a limit where this becomes like two infinite planes I think that's where you're thinking of and then in that case I'm going to get contributions from both equally and then I get net zero that's a really good line of reasoning that I had not considered before hmm I'm gonna think about that okay all right yes that's a good that's a that's a good point okay yes so let's stick to the to the sphere equations and what we know is that if this is an insulating sphere on the outside its contribution to the electric field of interior to itself is zero so if I'm at an interior point to that sphere I may as well throw out the outer spheres really so wherever I'm standing any spheres that are external to me throw them out now finish the problem and now I just get a contribution due to the Inner Sphere and that's going to give me answer be that in this middle region all I notice electric field wise is the contribution from the Inner Sphere so then the answer has to be B so we did our two spherical shells here's the answer is that interior to both of them I get no contribution in between them I only see the contribution of the interior sphere and then outside of them I see the contribution from both spheres okay now if I think then of adding up a bunch of spheres now I want to build up the the net contributions of a spherical ball okay so I'm going to take now the case where it's not a spherical shell but I want a ball of charge a whole sphere where the inside has charge inside of it as well so I'll distribute the charge all throughout the interior still think of an insulating sphere and I want to know what's the electric field outside and what's the electric field at various places inside that sphere so if I think about now what happens if I add up eight concentric spheres I'll just get a contribution from all eight of them if I'm at an observation point outside okay so I want to build up the solid sphere let me have a total charge on the solid sphere of total charge big Q its total volume is 4/3 PI R cubed okay and then I can also think about well what's the charge density density is always with respect to either volume or area or linear density it depends on your problem in this case I've uniformly distributed the charge over a volume so I need to think in terms of charge per volume so the charge per volume the total charge Q divided by the total volume 4/3 PI R cubed all right and all that was not confusing I hope now let me think about building up the total volume by summing up all these shells I want to sum up all the shells so let's stop at a point Midway let me say I'm going to make an observation point here at little R if I'm at little R then I feel a contribution from that middle sphere I feel a contribution from the next shell and the next shell and the next shell and the next shell okay all of those shells contribute to an electric field that I can feel at position little R inside but all these other shells are exterior and I don't notice the exterior shells so we just need to figure out if we can calculate what's the net electric field due to that inner ball then we know what the answer is so what if we stopped here at little R okay we're going to assume that the charge is distributed uniformly throughout the entire interior of the sphere so I just need to charge so far right if I want the electric field right here it only depends on the interior spherical shells how much charge do I have so far I don't know but we're gonna figure it out so I'm gonna write down the charge so far as Delta Q what's the volume so far well I do know the volume so far that's 4/3 PI little R cubed so it's my observation point cubed so I know the volume so far now here's the crux right I've uniformly distributed the charge throughout the interior of the sphere so any chunk I take out of the sphere let me just take a little chunk out of it I count the charges on it and if I also measure the volume and take the charge divided by the volume of that little chunk it's a number which is the same throughout the sphere so any chunk I take out I count the charge I divide by the volume I get the same number charge per volume is the same anywhere in the sphere so that means the charge for volume on my inner spheres right if I'm just talking about the red contribution so far the charge for volume is the charge so far divided by the volume so far right and whatever that is it has to equal the total charge divided by the total volume because charge provide is the same no matter where I look in the sphere so that's the key to solving this is that this charge for volume is the same as the charge so far divided by the volume so far you have a question about that step I'm just going to say this is equal to that and even though I didn't know how much the charge was so far I can now figure it out right if this is equal to that I can figure it out so here's what we found on the previous slide that the total charge for total volume is big Q divided by 4/3 PI big R cubed and what I have so far is the charge so far divided by the volume so far is the same so now I can solve for Delta Q which is the charge so far I just take this 4/3 pi little R cubed and move it to the other side and I get that it's big Q times this ratio of volumes the 4/3 cancels the pi cancels and I'm left with little R cubed divided by big R cubed and so then the electric field so far well from outside of the red spheres at this observation point little R the electric field so far just depends on how much charge I have so far so I put in the the equation then for outside of a sphere is 1 over 4 pie epsilon-not I need to charge so far divided by R squared and the charge so far was that Delta Q okay and Delta Q we said is big Q times little R cubed divided by big R cubed and little R cube divided by R squared gives me a net little R the one over big R cubed stays put and so this is the solution for inside of a solid sphere so the electric field inside of a solid sphere has this funny shape right my observation point is little R big R is the radius of the sphere and so if I know the total charge on the sphere then the electric field goes up linearly with my observation point because as I go out and out I'm seeing more and more spheres right I step backward and now there's another sphere inside I step backward and now there's another sphere inside so the electric field is getting stronger and stronger until when I get to the outside now it starts to fall off again like 1 over R squared okay so that's the summary then of the solid sphere inside the interior of the solid sphere the electric field goes up like are your observation point until you get to the outside and then it starts falling off like 1 over R squared okay all right we're done a little bit early today that's it for today so I'll see you guys next week

Channel: Prof. Carlson

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Length: 38min 3sec (2283 seconds)

Published: Thu Sep 29 2016