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# Magnitude of electric field created by a charge

## Video transcript

okay so we know that electric charges create electric fields and we know the definition of the electric field is the amount of force per charge what charge some charge that finds its way into this region let's say this charge right here so if we took the force on this charge let's give this a name let's call this q2 so we can keep these all straight and I'll call it q2 up here if we took the force on this charge Q 2 divided by Q 2 that would be the electric field at that point in space but something that would be useful to have is a formula that would let us figure out what's the electric field being created at that point in space without even referring to Q 2 so it'd be useful to have a formula that would let us figure out what's the electric field that this Q 1 is creating over here at this point in space without even referring to Q 2 at all so is there a formula for that there is and it's not that hard to find but the first thing I'll caution you about is that the formula we're about to find here is going to be for the magnitude of the electric field so I'm going to erase these vector crowns on these variables this formula we get will just be for the magnitude of the electric field and I'll tell you why in a second so the way we'll find a formula for the magnitude of the electric field is simply by inserting what we already know is the formula for the electric force so Coulomb's law gives us the force between two charges and we're just going to put that right in here Coulomb's law says that the electric force between two charges is going to be K the electric constant which is always 9 times 10 to the 9th multiplied by Q 1 the first charge that's interacting and that'd be this Q 1 over here multiplied by Q 2 the other charge interacting divided by the center-to-center distance between them squared and then because we're finding electric field in here we're dividing by Q 2 so notice what happens here Q 2 is cancelling and we get that the magnitude of the electric field is going to be equal to K this electric constant and I'll write that down over here so we know what it is K is 9 times 10 to the 9th and it's got kind of weird units but it makes sure that all the units come out okay when you multiply and then what do we still have up here we've still got a Q 1 divided by the center to Center distance between those two charges squared and you might be like well the other charge went away we cancelled it out centered between which two charges well this could be to any point in space really so you imagine your charge at any point you want I can put it here I can move it over to here the are would just be the distance from the first charge q1 to wherever I want to figure out what the electric field would be but since this q2 always divides out we don't even need to talk about that we can just figure out the electric field that's created by q1 at any point in space so this R is just the distance from the centre of the charge creating the field to the point in space where you want to determine the electric field and now we've got it this is a formula for the electric field created by a charge q1 technically though this is only true if this is a point charge in other words if it's really really small compared to the other dimensions in the problem or if this is spherically symmetric then it doesn't matter if you're outside of this charge and you've got a spherically symmetrical charge distribution where all the charges and lumped on one side of this sphere or anything like that if it's evenly distributed throughout then this formula also works just as well when you're outside the sphere and what's this formula saying it's saying that the absolute value or the magnitude of the electric field created at a point in space is equal to K the electric constant times the charge creating the field this is important this charge q1 is creating this electric field and then you plug in the distance away from that charge that you want to determine the electric field or you square it and that'll tell you what the magnitude of the electric field is created by q1 at any point in space around it now why are we being so careful saying that this is just the magnitude here's why imagine we plugged in this charge is positive because the charge creating it is positive you'd get a positive value for the electric field and you might think oh that means positive that means to the right and in this case it works out it does go to the right at this point but let's say you've put those same calculations for a point over here and you want to determine what's the value of the electric field at this point well if you plugged in K it's a positive number your Q is a positive number R is going to be a positive number even if you tried to plug in R as a negative it square away that negative anyway this would all come out positive so you might think Oh over here points to the right as well because that's the positive direction right well no this formula is not going to tell you whether this electric field goes up down left or right if you really wanted to include a positive or negative sign for this charge all that positive would be telling you is that the fields pointing radially away from the charge but radially away could mean left it could mean up it could mean right or it could mean down and all of those count as a positive as far as this formula would be concerned so that's why we say be careful you're just finding the magnitude when you find this formula the way you find the direction is just by knowing that the field created by a positive is always radially away from that positive but even though this formula just gives you the magnitude that's still really useful so we're going to use this this gives you the magnitude of the electric field from a point charge at any point away from that point charge so let's solve some examples here let's use this thing so let's say you had a positive two nano Coulomb charge and you wanted to determine the size and direction of the electric field at a point three meters below that charge so we want to know what's the size and direction of the electric field right there well to get the size we could use the new formula we've got which says that the electric field created by a charge Q is going to be equal to K times that Q over R squared so we'll use that down here so we get K which is always nine times ten to the ninth and then we multiplied by the charge creating the field which in this case it's this positive two nano Coulomb charge nano stands for 10 to the negative ninth and then we divided by the distance from our charge to the point where we want to find the field that's three meters you can't forget to square this people forget to square this all the time it doesn't come out right so you got to remember to square the R and if we solve this 10 to the ninth times 10 to the negative ninth goes away that's pretty nice and then nine divided by three squared is just nine divided by nine so all that's going to go away and all we really get for the electric field is that it's going to be two Newton's per Coulomb at this point here so that is the magnitude this gives us the magnitude of the electric field at this point in space that's how you get the size of it how do we get the direction we just asked what was creating this field it was a positive charge positive charges always create fuels that point radially away from them and at this point radially away from this positive is going to point straight down we've got an electric field from this to nano Coulomb charge the points straight down and has a value of two Newton's per Coulomb what did this number mean it means if we put another charge at that point in space some little charge q then there would be two newtons for every Coulomb of charge that you put there since we know that the electric field is the amount of force per charge notice that even though this electric field came out to be positive it was pointing down because all we're getting out of this calculation is the magnitude of the electric field let's do one more let's try this one out let's say you had a negative 4 micro Coulomb charge and you wanted to determine the size and direction of the electric field at a point 6 meters away to the left of that charge so we use the same formula we'll say that the electric field created by this negative charge is going to equal K which is always nine times ten to the ninth and then multiplied by the charge creating that field which in this case is negative four microcoulombs but I am NOT going to plug in this negative sign because I know all this formula is giving me is the magnitude of the electric field I'm not going to get tricked into thinking that this negative sign would tell me the direction of the electric field I mean it does tell you the direction it tells you that it points radially inward but it's safest got to take my word on this safest to just leave that negative sign out and know that this is just the magnitude of the electric field so we have four microcoulombs micro stands for 10 to the negative sixth and then we divide by the distance from the center of that charge to the point we want to determine the electric field out which is right here and we square it that's six meters and we cannot forget to square so if we solve this for the electric field we're going to get well 6 squared is 36 and 9 over 36 is 1/4 1/4 of 4 is just 1 so all were left with is 10 to the 9th times 10 to the negative sixth but that's just 10 to the 3rd which is 1,000 so this electric field is going to be 1,000 Newton's per Coulomb at that point in space that's the magnitude of the electric field or in other words that's the size of the electric field at that point how do we get the direction we're going to decide this by thinking carefully about it in other words we don't include this negative not because Direction isn't important we don't include this negative because Direction is so important we're going to make sure we get this right so what I say is that I've got a negative charge and I know that negative charges create fields that point radially into them and that means over here at this point to the left the electric field is pointing radially toward that negative charge it would point to the right so we have a thousand Newton's per Coulomb of electric field and would it would point to the right note if I would have just naively plugged this negative sign in over to here I would have come out with a negative value for my electric field and I might have thought well negative that means the leftward right so that means it points to the left and I would got the wrong direction so that's why we don't do that all this negative sign is representing if you were to plug it in is that it's pointing radially inward but radially in could mean right if you're over here to the left it could mean left if you're over here to the right it could mean up if you're underneath the charge and could mean down if you're above the charge so in other words it kind of doesn't mean anything for a given particular problem it often just screws you up so leave that out of there don't put the negative signs in just use this formula to get the magnitude and once you have that magnitude just know which direction negative charges create their fields and that'll tell you which direction the field points so recapping this is the formula for the electric field created by a charge Q and it tells you that the magnitude of the electric field is going to be equal to K the electric constant times the charge creating that field divided by the distance from the center of that charge to the point where you want to find the field squared