 ## Subtitles section Play video

• - [Instructor] 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,

• i.e. the total electric field,

• created halfway between these two charges down here.

• So you've got a positive eight nanocoulomb charge

• and a negative eight nanocoulomb 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 at first you might think, well you should

• just get zero, right?

• It's very tempting to say that the electric field

• is just gonna be zero there

• because you've got a positive eight nanocoulomb charge

• and a negative eight nanocoulomb charge

• and those should just cancel, right?

• But you have to be really careful,

• turns out that's not true here,

• this is not gonna 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 nanocoulomb charge is gonna

• create a field at this point that goes radially away

• from the positive charge, and so it's gonna go to the right.

• And I'm not even looking, so when 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.

• Then I just ask what field would this

• positive charge create?

• It's still gonna create that field

• whether this negative charge is over here or not.

• And now I can do the same thing, I can ask

• what field would this negative charge create?

• And I'm gonna pretend like this positive charge

• isn't even here.

• So negative charges create a field that 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,

• and that was to the right.

• So not only are these not gonna cancel,

• these are gonna add up to twice the fields

• cuz you're gonna add up these vectors,

• you just add them up if they're in the same direction,

• and you'll get two 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 a positive charge have to cancel

• their electric fields.

• Those electric fields might point the same direction,

• so you gotta be careful.

• So how do we find this net electric field then,

• what do we do?

• Well we're gonna say that, all right, this electric field,

• the first thing I can say is this net electric field

• is just gonna 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 gonna be equal

• to the sum of the electric fields

• each charge creates there.

• So we'll do the blue charge first, that's gonna be k

• times the blue charge divided by r squared.

• Then we'll do the yellow charge, it's gonna 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 10 to the ninth,

• and the q of this blue charge was positive eight

• nanocoulombs, 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 wanna 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,

• but that's not what I'm finding, I'm finding the field

• each charge creates at this halfway point.

• So I'm gonna plug in three meters down here,

• and I can't forget to square it.

• And now I have to be careful, just cuz my charge is positive

• doesn't necessarily mean that the contribution

• to the electric field is positive.

• You have to check, you can't rely on the sign

• of this charge to tell you whether the contribution's

• positive or negative.

• I've gotta look at what direction it 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.

• Then we're gonna have another term.

• I'm gonna leave off the plus or minus cuz, I mean,

• it might be plus, it might be minus,

• we'll leave that off for a second,

• we'll have to decide when we know what direction it goes.

• So we do nine times 10 to the ninth,

• and then the charge is negative eight nanocoulombs,

• but I am not gonna 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 find the field,

• and 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 gotta look at what direction it goes.

• Since it goes to the right, that's the positive direction,

• so this is gonna be plus, these add up,

• these both go the same direction, the positive direction,

• so the total net electric field is just gonna be

• both of these added up.

• So if I do this, if I square this three I'm getting nine,

• and nine divided by nine is just one,

• so I get eight newtons per coulomb,

• and then this term is really the same thing,

• nine is divided by nine so that goes away,

• 10 to the ninth cancels with 10 to the negative ninth

• and all I'm left with is this eight,

• so it'd be plus eight newtons per coulomb.

• So each charge is contributing eight newtons per coulomb

• of electric field at this point

• which means that the total net electric field

• would just be 16 newtons 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 cuz both of these vectors

• pointed to the right so the total is gonna 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 gonna be equal,

• you can just write one of them down and multiply by two,

• you don't have to just add them both up,

• but I wanted to show you this way so you could see

• how everything works out.

• And in the end we get 16 newtons per coulomb

• for the total field which points to the right.

• Now what if we changed this, what if we made this

• instead of a negative eight nanocoulomb charge

• we made this a positive eight nanocoulomb 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