Coulomb's Law: Calculating Electric Field Due to Multiple Charges

[flyingpig]

Homework Statement

[PLAIN]http://img547.imageshack.us/img547/7932/14207376.png

[PLAIN]http://img684.imageshack.us/img684/719/75458442.png

[PLAIN]http://img88.imageshack.us/img88/4883/55224430.png

Assume q_{1}=q_{2}=q_{3} and that all charges are positive.

The Attempt at a Solution

*if someone could, please tell me the proper code for vectors, because I am having trouble

For the first of the problem

\vec{E_{1}} = \vec{E_{21}}

Since it sort of just "sits in space", I put q_{2} on the origin.

So \vec{E_{21}} = <0, k\frac{q_2}{d^2}> and the magnitude should simply be k\frac{q_2}{d^2}.

For the second part

\vec{E_{1}} = \vec{E_{21}} + \vec{E_{31}}

\vec{E_{31}} = k\frac{q_{3}}{d^2}<-\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}>

Since \vec{E_{21}} = <0, k\frac{q_2}{d^2}>

Then the sum would be \vec{E_{1}}= \frac{k}{d^2}<-q_{3}\frac{\sqrt{2}}{2}, q_{3}\frac{\sqrt{2}}{2} + q_{2}>

 

[tiny-tim]

hi flyingpig!

(it's easiest to use bold letters for vectors )

i] the field is kq/d², but the force is kqq/d²

ii] i think you're using 45° for the angle, it's 60°

iii] don't forget the question says you can assume the qs are all the same!

 

[flyingpig]

hi flyingpig!

(it's easiest to use bold letters for vectors )

i] the field is kq/d², but the force is kqq/d²

ii] i think you're using 45° for the angle, it's 60°

iii] don't forget the question says you can assume the qs are all the same!

Oh right...what am I doing!? I was reading "electrostatic" and it immediately turned into field.

But what if I want to use arrows? I actually find it more difficult to tell if it is bold.

\vec{F_{1}} = \vec{F_{21}}

\vec{F_{21}}= k\frac{q^2}{d^2}<0,1>

That's part one

For part two

\vec{F_{1}} = \vec{F_{21}} + \vec{F_{31}}

\vec{F_{31}}= k\frac{q^2}{d^2}<-\frac{\sqrt{3}}{2}, \frac{1}{2}>

\vec{F_{21}}= k\frac{q^2}{d^2}<0,1>

So the sum of \vec{F_{21}} + \vec{F_{31}} is then

k\frac{q^2}{d^2}<-\frac{\sqrt{3}}{2}, \frac{3}{2}>

So now, my question is, should I turn this into a unit vector?

 

[cupid.callin]

Oh right...what am I doing!? I was reading "electrostatic" and it immediately turned into field.

Try to be polite next time.

______________________________________________________


No, why would turn this into a unit vector?
Unit vector has mag of 1
it will mean that that your force also has a unit vector of one!

 

[flyingpig]

Try to be polite next time.

______________________________________________________


No, why would turn this into a unit vector?
Unit vector has mag of 1
it will mean that that your force also has a unit vector of one!

I wasn't being rude =(! Not sure where you caught that from.

But I thought the unit vector will only take care of the direction of my force, not the force itself? Is it okay to leave my answer above like that?

Did I get it right!??

Thanks!

 

[cupid.callin]

Yes your answer is right.

And direction can still be founded with this result. unit vectors just provide direction without changing mag of something!

10(i) + 10(j) has same direction as its unit vector (1/√2)(i) + (1/√2)(j)

 

[flyingpig]

Yes your answer is right.

And direction can still be founded with this result. unit vectors just provide direction without changing mag of something!

10(i) + 10(j) has same direction as its unit vector (1/√2)(i) + (1/√2)(j)

I noticed something, this problem just want to assume that I place q₂ at the origin, why?

Also, I thought the concept of unit vector is like multiplying and dividing by one, does it really matter?

 

[cupid.callin]

I noticed something, this problem just want to assume that I place q₂ at the origin, why?

Well this problem doesnot depend on the choice of origin
Also, as long as orientation of XYZ is same direction will also remain same ... but mag of force always remain same no matter where is the origin

 

[tiny-tim]

Try to be polite next time.

cupid.callin, whatever are you talking about?

… So now, my question is, should I turn this into a unit vector?

Sort of.

The question asks for "the magnitude and direction".

So you need to find the magnitude anyway …

once you've done that, what's left is the unit vector of the direction!

I noticed something, this problem just want to assume that I place q₂ at the origin, why?

Also, I thought the concept of unit vector is like multiplying and dividing by one, does it really matter?

hmmm … not sure what you mean by either of those.

 

[flyingpig]

The question asks for "the magnitude and direction".

So you need to find the magnitude anyway …

once you've done that, what's left is the unit vector of the direction!

But I thought taking the unit vector according to cep will yield a magnitude of 1N

hmmm … not sure what you mean by either of those.

Sorry I wasn't sure what my question is now that I read over again.

 

[cupid.callin]

once you've done that, what's left is the unit vector of the direction!

Both unit vector and vector itself gives same direction.
If answer is written with mag <not original vector> then only you need to give unit vector, or even better just give answer in degrees!

If he's giving answer in original vector, why waste time in finding unit vector also?

 

[tiny-tim]

But I thought taking the unit vector according to cep will yield a magnitude of 1N

now I'm really confused …

you got kq²/x² < -√3/2, 3/2 > …

where does 1N come into that?

you need to convert < -√3/2, 3/2 > to a magnitude times a unit vector

Sorry I wasn't sure what my question is now that I read over again.

he he

 

[cupid.callin]

you got kq²/x² < -√3/2, 3/2 > …

where does 1N come into that?

you need to convert < -√3/2, 3/2 > to a magnitude times a unit vector

you agree that unit vector of < -√3/2, 3/2 > will have mag of 1, right?

So by this you mean that net force is just kq²/x²

< -√3/2, 3/2 > has nothing to do in the mag?

 

[flyingpig]

now I'm really confused …

you got kq²/x² < -√3/2, 3/2 > …

where does 1N come into that?

you need to convert < -√3/2, 3/2 > to a magnitude times a unit vector


he he

I got it from this

No, why would turn this into a unit vector?
Unit vector has mag of 1
it will mean that that your force also has a unit vector of one!

But once I get it to the unit vector, wouldn't my force be one like he said? Or is that actually not surprising since the distance is is d after all?

 

[cupid.callin]

But once I get it to the unit vector, wouldn't my force be one like he said? Or is that actually not surprising since the distance is is d after all?

Which distance is d?

 

[tiny-tim]

you agree that unit vector of < -√3/2, 3/2 > will have mag of 1, right?

So by this you mean that net force is just kq²/x²

< -√3/2, 3/2 > has nothing to do in the mag?

cupid.callin, are you drunk?

flyingpig, please ignore his advice.

But once I get it to the unit vector, wouldn't my force be one like he said? Or is that actually not surprising since the distance is is d after all?

express < -√3/2, 3/2 > as a magnitude times a unit vector, say M <a,b> where <a,b> is the unit vector

then the magnitude of the force is M times kq²/x², and it is in the direction of <a,b>

 

[flyingpig]

Which distance is d?

We don't know! It could be anything and it shouldn't matter

 

[flyingpig]

cupid.callin, are you drunk?

flyingpig, please ignore his advice.


express < -√3/2, 3/2 > as a magnitude times a unit vector, say M <a,b> where <a,b> is the unit vector

then the magnitude of the force is M times kq²/x², and it is in the direction of <a,b>

Don't you mean 1/M?

k\frac{q^2}{\sqrt{3}{d^2}}&lt;-\frac{\sqrt{3}}{2}, \frac{3}{2}&gt;

But then I would get back at k\frac{q^2}{d^2}

 

[tiny-tim]

hi flyingpig!

(just got up :zzz: …)

no …

for example if the vector was 3 <4,0>,

that would be 12 <1,0> …

magnitude 12, in the direction of unit vector <1,0>

 

[flyingpig]

Then what am I thinking of? For 3<4,0>

M = 12

So 1/12<12,0>?

 

[tiny-tim]

1/12 of <12,0> is the unit vector <1,0> …

are you thinking of a way to make unit vectors? ​

 

[flyingpig]

1/12 of <12,0> is the unit vector <1,0> …

are you thinking of a way to make unit vectors? ​

Isn't that the goal?

 

[tiny-tim]

half the goal …

the goal (in the original question) was to find the magnitude and the direction …

ie a magnitude and a unit vector

 

[flyingpig]

half the goal …

the goal (in the original question) was to find the magnitude and the direction …

ie a magnitude and a unit vector

k\frac{q^2}{\sqrt{3}{d^2}}&lt;-\frac{\sqrt{3}}{2}, \frac{3}{2}&gt; <=== Unit vector

k\frac{\sqrt{3}q^2}{d^2} <=== magnitude.

 

[tiny-tim]

k\frac{q^2}{\sqrt{3}{d^2}}&lt;-\frac{\sqrt{3}}{2}, \frac{3}{2}&gt; <=== Unit vector

k\frac{\sqrt{3}q^2}{d^2} <=== magnitude.

<-√3/2, 3/2> is not a unit vector

 

[flyingpig]

<-√3/2, 3/2> is not a unit vector

But I had \sqrt{3} in the denominator.

 

[tiny-tim]

i don't understand …

are you saying that <-√3/2, 3/2> is a unit vector?

 

[flyingpig]

i don't understand …

are you saying that <-√3/2, 3/2> is a unit vector?

No, but 1/√3<-√3/2,3/2> is

 

[tiny-tim]

No, but 1/√3<-√3/2,3/2> is

yeeees … why not just write it < -1/2, √3/2 > ?

(btw, you could have seen, just by looking at the original diagram, that the angle was going to be 30° )

 

[flyingpig]

yeeees … why not just write it < -1/2, √3/2 > ?

(btw, you could have seen, just by looking at the original diagram, that the angle was going to be 30° )

That's what I had here


k\frac{q^2}{\sqrt{3}{d^2}}&lt;-\frac{\sqrt{3}}{2}, \frac{3}{2}&gt;

But when I take the magnitude, it will be one, not √3 of the force.

 

[tiny-tim]

we seem to be completely misunderstanding each other

i'm saying that < -√3/2, 3/2 > is not a unit vector, so it shouldn't be part of the answer …

why do you keep writing it? what is it for?

 

[flyingpig]

we seem to be completely misunderstanding each other

i'm saying that < -√3/2, 3/2 > is not a unit vector, so it shouldn't be part of the answer …

why do you keep writing it? what is it for?

But 1/√3<-√3/2, 3/2> is

I had a √3 beside a d^2

Do you see?

 

[tiny-tim]

yeees …

but, as i said, why do you keep writing < -√3/2, 3/2 > as part of the final answer?

it's not a unit vector, so what is it there for?

 

[flyingpig]

yeees …

but, as i said, why do you keep writing < -√3/2, 3/2 > as part of the final answer?

it's not a unit vector, so what is it there for?

But it isn't <-√3/2, 3/2>, it is 1/√3<-√3/2,3/2>

 

[flyingpig]

Are we going in circles?

 

[tiny-tim]

Are we going in circles?

yes … isn't that the flyingpig's natural method of travel?

i can't force you to write a unit vector!

 

[flyingpig]

yes … isn't that the flyingpig's natural method of travel?

i can't force you to write a unit vector!

This is bugging me, if I switch it to i and j notations

Namely, my final solution giving me 1.6N(-√3/2i + 3/2j) isnot a unit vector, but if 1.6N/√3(-√3/2i + 3/2j) then is a unit vector. So how do I deal with the magnitude? Should I take √3(1.6N) or a magnitude of one from my unit vector?

 

[tiny-tim]

the way to convert 1.6(-√3/2i + 3/2j) into a magnitude and a unit vector is:

1.6 √3 (-1/2i + √3/2j) ​

 

[flyingpig]

the way to convert 1.6(-√3/2i + 3/2j) into a magnitude and a unit vector is:

1.6 √3 (-1/2i + √3/2j) ​

You multiply and divided the vector by its magnitude, is that what you did? If so, then what was I doing? I can see you destributed the 1/√3 into the unit vector, but you left the top √3 beside 1.6N

 

[flyingpig]

Let me add another question to clarify the question I just asked.

Isn't 1/√3(-√3/2i + 3/2j) a unit vector? At least in my Linear Algebra Book

 

[tiny-tim]

You multiply and divided the vector by its magnitude, is that what you did? If so, then what was I doing? I can see you destributed the 1/√3 into the unit vector, but you left the top √3 beside 1.6N

i only left the √3 separate because i didn't want to bother to calculate 1.6 √3 !

Let me add another question to clarify the question I just asked.

Isn't 1/√3(-√3/2i + 3/2j) a unit vector? At least in my Linear Algebra Book

yes of course …

but it's a very strange way of writing it, when (-1/2i + √3/2j) is so much simpler, and you'll certainly lose a mark for it in the exam

 

[flyingpig]

i only left the √3 separate because i didn't want to bother to calculate 1.6 √3 !


yes of course …

but it's a very strange way of writing it, when (-1/2i + √3/2j) is so much simpler, and you'll certainly lose a mark for it in the exam

Why is that (-1/2i + √3/2j) simpler? If we are just talking about numbers here where 1.6 is a scalar (talking only about the numbers for now) then, 1.6/√3 <-√3/2, 3/2> is a unit vector, but that is not the same as 1.6√3<-1/2, √3/2>.

(sorry for switching notations all of a sudden, I have no idea what i was thinking)

In 1.6/√3 <-√3/2, 3/2>, I needed (wel, you needed) to multiply √3 again to get 1.6√3<-1/2, √3/2>

But noting that 1.6√3 is a scalar that will EXTEND the unit vector<-1/2, √3/2>, so how is that different?

 

[tiny-tim]

hi flyingpig!

Why is that (-1/2i + √3/2j) simpler?

because it's a unit vector!

If we are just talking about numbers here where 1.6 is a scalar (talking only about the numbers for now) then, 1.6/√3 <-√3/2, 3/2> is a unit vector

no it isn't!

 

[flyingpig]

hi flyingpig!


because it's a unit vector!


no it isn't!

I think I got confused with the outside scalar being inside the vector itself. In that case, why don't I just distribute the 1.6N?

 

[flyingpig]

My book says "Leave your answers in F_xi + F_yj

 

[tiny-tim]

My book says "Leave your answers in F_xi + F_yj

No, it doesn't! …

in post #1, you provided a photo of the book, it clearly says "What is the magnitude and direction … Leave your answer in algebraic form" …

you need to state the magnitude and direction !

 

[flyingpig]

Never mind, if i need to use Fxi and Fyj I will just convert it to a unit vector, that's all I really need to do.