Charge density & continuous charge distribution

let's talk about charge distributions charge distribution basically means collection of charges so it is collection of charges and you've actually dealt with them for example you may have dealt with situations where you were given there is a i don't know maybe a plus one nanocoulomb chart somewhere and there's a minus five nanocoulomb chart somewhere and then you are asked to do maybe calculate the force between them or asked to calculate electric field somewhere so this is a collection of charge and so this is a charge distribution but you know this particular kind of charge distribution is called discrete charge distribution what do i mean by that what i mean is here you can count and you can say ah there is one charge over here and there is another charge over here and there are no charges in between when you can count and you have such point charges we call that collection a discrete charge distribution but the focus of this video is the other kind which i find more interesting this is called continuous charge distribution continuous and let me immediately give you an example of that so let's say we have a wire and let's imagine i charge up this wire and let's say it gets a positive charge so now the positive charge will be continuously spread along this wire now i don't mean that there is a charge here and then there are no charges in between and there is a charge here that's not what i'm saying what i'm saying now is the charges are spread out there's charge everywhere this is what we call a continuous charge distribution and do you know what's interesting about this i can have that same wire and i can have that same amount of charge let's say this total charge is i don't know maybe 100 coulomb and i can distribute it continuously in a different way for example you know what i how can do i can i can distribute it maybe somewhat like this now again i don't mean that there are no charges in between it's a continuous distribution there are charges everywhere and so the total charge here is the same as the total charge over here but what's the difference well the difference is the charges are more crowded over here and they are less crowded over here that's what i'm representing but over here the charges are equally crowded everywhere and that's why when we're talking about continuous charge distributions this is an important quantity how crowded the charges are are they uniformly crowded everywhere or are they more covered in one place and less crowded at some other place and that's why when it comes to continuous charge distribution we introduce a new quantity called charge density and you may have heard of the word density the word density sorry the word density basically means how crowded something is so when we say charge density we're talking about how crowded the charges are and then we're dealing with uh you know charge distribution over a line and we'll talk more about different kinds of distribution in a minute whenever we're dealing with that thus the symbol that we use to represent charge density is a greek symbol lambda and this number basically tells you how crowded the charges are on a given length and so the way we represent lambda or with the way we calculate lambda is we say how much charge is present over unit length and i'll give you an example but before that let's talk about the units what will be the units of charge density well the unit of charge density would be coulomb per meter so for example i could say hey the charge density over here for this this wire is 20 coulomb per meter what does that mean that means if i were to take one meter of this wire and i don't know how long this wire is but let's say it's a very very long wire and i took one meter of this in that one meter i would find 20 coulombs of charge that's what this means and i can take that one meter anywhere and i'll find 20 coulombs the charge density over here is a constant because it's equally crowded everywhere so what if i take 2 meters of length how much charge would i find over there well per meter is 20 so in two meters i'll find 40 coulombs of charge if i take half a meter how much charge would i find over there well i would find 10 coulombs of charge does this make sense do you understand what charge density is okay what about charge density over here can i give one number for charge density over here no because the charge density varies you have a you can kind of say there's a very high charge density over here there is very low charge density over here and how do i represent that how do i figure how do i give a number to it the way to give a number over here is you could say you could take a very tiny very tiny length and let's say that length is i don't know maybe a nanometer okay take a very very tiny length and in that length find out how many charges are present maybe in that one nanometer i find i don't know maybe 50 nano coulombs of charge now i will say at that point lambda which is our charge density is 50 nano coulombs divide by one nanometer so that would be 50 coulomb per meter and now it's interesting is this does not mean i take one meter and i'll find 50 coulombs no this basically says at that point you will find if you take the charge and divide by length that ratio turns out to be 50. so this is the value at that point similarly if i were to take maybe at this point i take one nanometer over here and i'm just using nano as an example you imagine it's a very incredibly tiny tiny point you know tiny distance you're taking in that one nanometer maybe i'll just find only two nano coulombs at this point so we will say the lambda at this point would be two nano coulomb divide by nanometer it will be two coulomb per meter so over here how do you calculate charge density well we calculate charge density lambda by taking very tiny lengths dl that's how you write it mathematically even though i've shown using like real numbers over here but mathematically it's an infinitesimal length you take and over that you find how much charge is present and that ratio tells you the charge density at individual points so if the charge is uniformly distributed then whether you take it at a point or you take it over a large length that value stays the same and that's pretty much about charge distribution but now you could ask wait a second here charges were distributed over a line and so we like to call this a continuous line charge distribution and therefore this density we like to give it a name we call this linear charge density because it's telling you how much charge is distributed over a line over a meter right so this is one kind of charge distribution are there other kinds of charge distributions the answer is yes and if you got this then rest of it's very similar so let me show you another kind in fact there are three kinds so the second one let me just write over here or let me do that over here in fact let me do that over here second one would be surface charge distribution surface charge distribution and you can kind of kind of imagine just by looking at the word now instead of charges being distributed over a line we have charges distributing over a surface so you can imagine let's say this is some kind of a i know maybe a metallic plate you can imagine this is a surface of a balloon whatever you want and let's imagine now there is there are charges everywhere on the surface and again it's continuous it charges everywhere this is what we call a surface charge distribution and we have the same nuances like we saw over here you could either have uh distributions which is uniform so the density would be the same everywhere or you could have non-uniform but my question is how would you define charge density here here it was charged per length coulomb per meter what would be over here what would be the units can you pause and think about it well over here since we're charges are distributing over an area here we introduce surface charge density and the symbol we use is sigma and surface charge density is just charge divided by area so the unit over here would be coulomb per meter square so for example if i said this was a uniform charge distribution and let's say sigma is i don't know maybe 10 coulomb per meter square what am i saying what is the meaning of that it means if i go anywhere on this surface and if i take let's say one meter square area i'll find 10 coulombs of charge on it but if i take 2 meter square area then i'll find 20 coulombs of charge on it and so on and so forth and lastly the third and the final kind of charge distribution i'm just like i've not arranged this very nicely but the third kind would be volume charge distribution and again i'm pretty sure you can kind of guess what this is now charges are distribution charges are distributed in space in in 3d space so you can kind of imagine let's say there is a box i am going to draw a cube type a cube over here or cuboid over here maybe this is my room yeah my room is cuboid i think okay anyways now imagine there are charges everywhere inside in 3d space this is volume charge distribution and exactly the same thing holds here you can either have uniform charge you know charges uniformly distribute over a volume or you can have them non-uniformly distributed over volume and again the question is how do you define a charge density over here can you think well since it's it's uh it's spread over a space here we define charge density as and the symbol we use is rho i think these symbols is something you need to remember but with the way we define it over here would be charge per volume and the units would be coulomb per meter cubed and just like before if you are dealing with non-uniform charge distribution then you have to do dq over dv at every point you would have different volume charge distributions so let's put it all together let's put this all in one screen let me all right so if you put it all together continuous charge distribution is basically when charges are spread out continuously you can have them spread out over a line or a surface or over a volume and one of the most important quantities is charge density it tells you how crowded charges are distributed over a line over a surface or over volume if the value is a constant which is my favorite then that means it's a uniform distribution if the value is not constant it changes at different different points then you'll have to use differential differentiations then it means the charge density charge distribution it's not uniform it's varying at some points you have more crowdedness compared to some other points