Net electric field from multiple charges in 1D

so it turns out solving electric field problems gets significantly harder when there's multiple charges I mean theoretically it shouldn't but people have a lot more problems when there's multiple charges involved so say the question is this let's say we wanted to know what's the magnitude and direction of the net electric field ie the total electric field created halfway between these two charges down here so you've got a positive eight nano Coulomb charge and a negative eight nano Coulomb charge and they're separated by six meters from the center to Center distance but what we want to know is what's the total electric field that they both create right there so each charge is going to create an electric field at this point and if you add up like vectors those electric fields what total electric field would you get now first you might think well you should just get zero right it's very tempting to say that the electric field is just going to be zero there because you got a positive eight nano Coulomb charge and a negative eight nano Coulomb charge and those should just cancel right but you have to be really careful that turns out that's not true here this is not going to be true and to see why first you should just draw what is the direction of each field at that point so this positive eight nano Coulomb charge is going to create a field at this point that goes radially away from the positive charge and so it's going to go to the right and I'm not even looking so what I'm trying to find the electric field from this positive charge over here I'm not even paying attention to this negative charge I pretend like this negative charge doesn't even exist and I just ask what field with this positive charge create it's still going to create that field whether this negative charge is over here or not and I can do the same thing I can ask what field with this negative charge create and I'm going to pretend like this positive charge isn't even here so negative charges create a field to go radially in so over here radially in would point to the right so these don't cancel the negative charge created a field radially in that was to the right the positive charge created a field radially out of the positive charge that was to the right so not only are these not going to cancel these are going to add up to twice the fields because you're going to add up these vectors you just add them up if they're in the same direction and you get 2 times the contribution from one of them so it's not always the case in other words it's not always the case that a negative charge and positive charge have to cancel their electric fields those electric fields might point the same direction so you got to be careful so how do we find this net electric field than what we do what we're going to say that alright this electric field the first thing I could say is this net electric field is just going to point in the X direction so this is just really in the X direction all I really care about is the electric field in this horizontal direction and it's going to be equal to the sum of the electric fields each charge creates there so we'll do the blue charge first that's going to be K times the blue charge divided by R squared then we'll do the yellow charge is going to be plus K the charge of that yellow charge divided by R squared so we'll plug in some values here this K is always nine times ten to the ninth and the Q of this blue charge was positive eight nano coulombs Nano is 10 to the negative ninth I like using Nano because then that negative nine cancels with that positive nine and what distance do I put in here a lot of people want to put in six but that's not what I want think about it I want the net electric field halfway between the two charges so the R that I care about in this electric field formula is the distance from the charge to the point where I want to determine the electric field and in that case this is three meters so for this case from the charge to the point I'm concerned about finding the field is three meters not six meters if we were finding the force these charges exert on each other then I'd have to use six meters that's not what I'm finding I'm finding the field each charge creates at this halfway point so I'm going to plug in three meters down here and I can't forget to square it and now I have to be careful that just because my charge is positive doesn't necessarily mean that the contribution to the electric field is positive you have to check you have to you can't rely on the sign of this charge to tell you whether the contribution is positive or negative I've got a legal what direction of points the direction this positive charge creates a field is to the right since that's typically the direction we call positive then I'm okay with calling this entire term here positive and we're gonna have another term I'm going to leave off the plus or minus because I mean it might be plus it might be - we'll leave that off for a second we'll have to decide when we know what direction goes we do nine times ten to the ninth and then the charge is negative eight nano coulombs but I am NOT going to plug in the negative sign oops and I left off Coulomb on the other one here sorry and then again the distance I want is from the charge to the point where we want to the field and noting that again is three meters and we can't forget to square it so should this contribution be positive or negative I can't rely on the negative sign to tell me that I've got a look at what direction it goes since it goes to the right that's the positive direction so there's going to be plus these add up these both go the same direction the positive direction so the total net electric field is just going to be both of these added up so if I do this if I square this 3 I get 9 + 9 divided by 9 is just 1 so I get 8 Newton's per Coulomb and then this term is really the same thing 9 is divided by 9 so that goes away 10 to the 9th cancels with 10 to the negative ninth and all I'm left with is this 8 so it'd be plus 8 Newton's per Coulomb so each charge is contributing 8 Newton's per Coulomb of electric field at this point which means that the total net electric field would just be 16 Newton's per Coulomb at that point that is the net electric field that's the magnitude of the net electric field at that point between them and which way does it go what's the direction it goes to the right because both of these vectors pointed to the right so the total is going to be twice as big as one of them and also to the right now if you have a case like this and both terms you know both terms are going to be equal you can just write one of them down and multiply by 2 you don't have to just add them both up but I wanted to show you this way so you can see how everything works out and in the end we get 16 Newton's per Coulomb for the total field which points to the right now what if we change this what if we made this instead of an negative 8 nano Coulomb charge we made this a positive 8 nano Coulomb charge well it would no longer create an electric field that points to the right positive charges create fields that point radially away from them so it would create its electric field to the left which means down here when we find its contribution to the electric field we'd have to include it as a negative contribution because it's pointing in the negative direction even though it's a positive charge the contribution it gives to the total electric field is negative because it points in the negative direction and that would give me zero so if I had this a positive this whole thing would add up to zero because I have eight and then minus eight and I'd get zero Newton's per Coulomb so the electric field would completely cancel right in the middle so what I'm saying is you have to be very careful with your negative signs don't just assume these contributions are always going to add up you can find each one always plugging in the charges is positive even if they're negative and then decide should I add or subtract these contributions based on whether they go to the right or to the left if they point to the right you choose a positive in front of this term since it points in the positive x-direction and if they point to the left you're going to choose a negative in front of this term because it would point in the negative x-direction so recapping define the total electric field from multiple charges draw the electric field each charge creates at the point where you want to determine the total electric field use this formula to get the magnitude of the contribution from each charge then decide whether those contributions should be positive or negative based not on the sign of the charge but the direction the field is pointing from that charge add up the two contributions and that will give you the total electric field at that point