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- [Instructor] So it turns
out solving electric field

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problems gets significantly harder

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when there's multiple charges.

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I mean, theoretically it shouldn't,

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but people have a lot more problems

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when there's multiple charges involved.

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So say the question is this;

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let's say we wanted to
know what's the magnitude

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and direction of the net electric field,

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i.e. the total electric field,

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created halfway between
these two charges down here.

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So you've got a positive
eight nanocoulomb charge

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and a negative eight nanocoulomb charge,

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and they're separated by six meters

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from the center to center distance.

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But what we want to know is
what's the total electric field

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that they both create right there?

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So each charge is going to
create an electric field

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at this point, and if
you add up like vectors,

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those electric fields,
what total electric field

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would you get?

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Now at first you might
think, well you should

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just get zero, right?

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It's very tempting to say
that the electric field

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is just gonna be zero there

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because you've got a positive
eight nanocoulomb charge

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and a negative eight nanocoulomb charge

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and those should just cancel, right?

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But you have to be really careful,

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turns out that's not true here,

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this is not gonna be true.

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And to see why, first you should just draw

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what is the direction of
each field at that point?

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So this positive eight
nanocoulomb charge is gonna

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create a field at this point
that goes radially away

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

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And I'm not even looking,
so when I'm trying to find

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the electric field from this
positive charge over here,

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I'm not even paying attention
to this negative charge,

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I pretend like this negative charge

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doesn't even exist.

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Then I just ask what field would this

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positive charge create?

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It's still gonna create that field

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whether this negative
charge is over here or not.

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

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what field would this
negative charge create?

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And I'm gonna pretend
like this positive charge

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isn't even here.

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So negative charges create
a field that go radially in.

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So over here radially in
would point to the right.

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So these don't cancel.

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The negative charge created
a field radially in,

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that was to the right, the
positive charge created a field

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radially out of the positive charge,

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and that was to the right.

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So not only are these not gonna cancel,

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these are gonna add up to twice the fields

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cuz you're gonna add up these vectors,

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

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and you'll get two times the contribution

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from one of them.

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So it's not always the
case, in other words

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it's not always the case
that a negative charge

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and a positive charge have to cancel

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their electric fields.

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Those electric fields might
point the same direction,

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so you gotta be careful.

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So how do we find this
net electric field then,

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what do we do?

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

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the first thing I can say
is this net electric field

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is just gonna point in the x direction.

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So this is just really in the x direction,

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all I really care about
is the electric field

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in this horizontal direction,
and it's gonna be equal

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to the sum of the electric fields

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each charge creates there.

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

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times the blue charge
divided by r squared.

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Then we'll do the yellow
charge, it's gonna be

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plus k, the charge of that yellow charge,

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divided by r squared.

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So we'll plug in some values
here, this k is always

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nine times 10 to the ninth,

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and the q of this blue
charge was positive eight

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nanocoulombs, nano is 10
to the negative ninth,

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I like using nano because
then that negative nine

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cancels with that positive nine.

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And what distance do I put in here?

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A lot of people wanna put in six,

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but that's not what I want.

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Think about it, I want
the net electric field

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halfway between the two charges,

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so the r that I care about in
this electric field formula

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is the distance from
the charge to the point

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where I want to determine
the electric field,

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and in that case this is three meters.

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So for this case, from
the charge to the point

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I'm concerned about finding the field

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is three meters, not six meters.

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If we were finding the
force these charges exert

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on each other, then I'd
have to use six meters,

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

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each charge creates at this halfway point.

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

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and I can't forget to square it.

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

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doesn't necessarily mean
that the contribution

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to the electric field is positive.

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

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of this charge to tell you
whether the contribution's

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positive or negative.

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I've gotta look at what
direction it points,

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the direction this positive
charge creates a field

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is to the right.

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Since that's typically the
direction we call positive,

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then I'm okay with calling
this entire term here positive.

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Then we're gonna have another term.

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

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it might be plus, it might be minus,

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we'll leave that off for a second,

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

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So we do nine times 10 to the ninth,

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and then the charge is
negative eight nanocoulombs,

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but I am not gonna plug
in the negative sign.

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Oops, and I left off coulomb
on the other one here, sorry.

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And then again, the distance
I want is from the charge

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to the point where we
want to find the field,

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and that again is three
meters, and we can't forget

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to square it.

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So should this contribution
be positive or negative?

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I can't rely on the negative
sign to tell me that,

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I've gotta look at what direction it goes.

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

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so this is gonna be plus, these add up,

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these both go the same direction,
the positive direction,

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so the total net electric
field is just gonna be

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both of these added up.

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

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and nine divided by nine is just one,

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so I get eight newtons per coulomb,

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and then this term is
really the same thing,

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nine is divided by nine so that goes away,

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10 to the ninth cancels with
10 to the negative ninth

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and all I'm left with is this eight,

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so it'd be plus eight newtons per coulomb.

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So each charge is contributing
eight newtons per coulomb

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of electric field at this point

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which means that the
total net electric field

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would just be 16 newtons
per coulomb at that point.

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That is the net electric
field, that's the magnitude

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of the net electric field
at that point between them.

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And which way does it
go, what's the direction?

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It goes to the right cuz
both of these vectors

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pointed to the right so
the total is gonna be

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twice as big as one of
them and also to the right.

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Now if you have a case
like this and both terms,

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you know both terms are gonna be equal,

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

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

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

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how everything works out.

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And in the end we get
16 newtons per coulomb

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for the total field which
points to the right.

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

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instead of a negative
eight nanocoulomb charge

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we made this a positive
eight nanocoulomb charge?

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Well it would no longer
create an electric field

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that points to the right.

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Positive charges create
fields that point radially

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away from them, so it would
create its electric field

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to the left, which means
down here when we find

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its contribution to the electric field

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we'd have to include it
as a negative contribution

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cuz it's pointing in
the negative direction.

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Even though it's a positive
charge, the contribution

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it gives to the total
electric field is negative

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cuz it points in the negative direction.

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And that would give me zero,
so if I had this a positive

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this whole thing would add
up to zero cuz I'd have eight

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and then minus eight and I'd get zero

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newtons per coulomb, so the electric field

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would completely cancel
right in the middle.

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So what I'm saying is you
have to be very careful

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with your negative
signs, don't just assume

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these contributions are
always gonna add up.

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You can find each one always
plugging in the charges

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as positive even if they're negative

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and then decide should I add or subtract

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these contributions based on whether

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they go to the right or to the left.

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If they point to the right
you'd choose a positive

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in front of this term since it points

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in the positive x direction.

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And if they point to the
left you're gonna choose

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a negative in front of this term

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because it would point in
the negative x direction.

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So recapping, to find
the total electric field

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from multiple charges,
draw the electric field

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each charge creates at the
point where you want to

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determine the total electric field,

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use this formula to get the
magnitude of the contribution

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from each charge, then decide
whether those contributions

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should be positive or
negative based not on the sign

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of the charge but the
direction the field is pointing

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from that charge, add up
the two contributions,

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and that'll give you
the total electric field

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at that point.