Position of a charged particle

[boxybrownmd]

Homework Statement

Two particles 1 and 2, each carrying 6.0 nC of charge, are located along an x axis, one particle atx=−30 mm and the other at x = 30 mm. Where along the y-axis is a particle 3 carrying a charge of +2.0 nC if it experiences an electric force of 6.9 × 10^−5 N ȷ^?

Homework Equations

Coulomb's law: F = (k*q1*q2)/d^2

Pythagorean Theorm: c = SQRT((a^2) + b^2))

The Attempt at a Solution

So I know the answer is 10mm and 40mm. However, throughout my work, I am not even close to said values. Here is the work I have done:

Some notes on my progress:

1.) I first began by finding Fx (since Fy was defined in the problem as 6.9x10^-5, and was split since I only need the force of a single 6.0 nC charge).
2.) After calculating this value, I took both force vectors and added them together to obtain the final total Force between particle 1 or 2 and particle 3.
3.) Taking this force value, I plugged it back into Coulomb's law in an attempt to obtain the value of d (In this problem, x = 30mm, y is what I am trying to find. So by finding d, I can use Pythagorean to calculate it). However... this is where things break apart. When I find d, I realize that this value is SMALLER than my x value... which makes 0 sense. From here I became absolutely lost.
https://scontent.fewr1-2.fna.fbcdn.net/v/t34.0-12/14237562_1253883344636041_2264484480519635381_n.jpg?oh=5f8d0bafacc2fd1b915de0e148810ab5&oe=57D91A9E
https://scontent.fewr1-2.fna.fbcdn.net/v/t34.0-12/14330114_1253883357969373_2960711846063879329_n.jpg?oh=a12feff8c85bc3b70ac88aebe43c94cb&oe=57D84E03

If someone could guide me from this point, I would greatly appreciate it! Thank you!

 

[kuruman]

Hi boxybrownmd and welcome to PF



I think you need to start over and draw a diagram of the three charges showing the forces on the third charge due to the other two charges. Then add these forces as vectors. In what direction do you think is the resultant?

 

[boxybrownmd]

I think you need to start over and draw a diagram of the three charges showing the forces on the third charge due to the other two charges. Then add these forces as vectors.

Well the issue is that I cannot draw it exactly since I do not know where the third particle technically would be?

 

[kuruman]

You know it's on the y-axis so draw it at some distance on the axis and label it y. Then draw your vectors.

 

[boxybrownmd]

https://scontent.fewr1-2.fna.fbcdn.net/v/t34.0-12/14344969_1253913204633055_3281124858787924149_n.jpg?oh=6d016711bbcdadc3abae4e72aef365df&oe=57D9162B
Edit: Forgot the picture :)

You know it's on the y-axis so draw it at some distance on the axis and label it y. Then draw your vectors.

I have attempted this but I am still lost on where to progress... I feel still at a loss on how to progress at this point. I mean, from a mathematical standpoint, its impossible for the overall Force value to have a smaller distance than x or y, but based on my math above, points towards that.

 

[kuruman]

Your drawing shows only two charges. There are three. Draw all three in the positions given by the problem. Then just draw the forces without calculating any numbers. This will help you visualize what's going on.

 

[boxybrownmd]

Your drawing shows only two charges. There are three. Draw all three in the positions given by the problem. Then just draw the forces without calculating any numbers. This will help you visualize what's going on.

I attempted to draw the diagram, but I do not understand anything else about the problem. I know that both of the charged particles (1 and 2) exert force on particle 3 in mirrored directions, which causes Fx to cancel but Fy to remain. However, in order to calculate F, you still need to know what Fx is?

 

[kuruman]

What do mean by Fx? If you mean the x-component of the net force on charge 3, that is zero. You said so yourself.

 

[kuruman]

Charges 1 and 2 exert separate forces on charge 3. Can you draw these two separate forces?

 

[boxybrownmd]

Charges 1 and 2 exert separate forces on charge 3. Can you draw these two separate forces?

Wouldn't the charges exerted on Particle 3 from 1 and 2 just be opposites but equal? For example, if Particle 1 exerted 1 N of force to the right, than particle 2 exerts 1 N to the left.

 

[kuruman]

Charge 1 exerts a force to the right and up. Charge 2 exerts a force to the left and up. The right and left cancel but the up and up add. In other words, the net force on charge 3 has zero x-component and non-zero y-component. It's the y-component of the net force that you are asked to find.

 

[boxybrownmd]

Charge 1 exerts a force to the right and up. Charge 2 exerts a force to the left and up. The right and left cancel but the up and up add. In other words, the net force on charge 3 has zero x-component and non-zero y-component. It's the y-component of the net force that you are asked to find.

From this point is where I am lost then...

 

[kuruman]

Like I said, draw a picture of the three charges on the axes as given by the problem. Then draw two arrows on charge 3 representing the forces from the other two charges. Find expressions, not numbers, for the y-components of these arrows and add them together. That will give you an expression for the net force on charge 3.

 

[boxybrownmd]

Like I said, draw a picture of the three charges on the axes as given by the problem. Then draw two arrows on charge 3 representing the forces from the other two charges. Find expressions, not numbers, for the y-components of these arrows and add them together. That will give you an expression for the net force on charge 3.

I feel like this does not help me find the answer... I still don't understand exactly what this will do for me. It just winds up being the original net force I started with and the x forces canceling...

 

[kuruman]

If you find an expression for the net force on charge 3, it will be a function of y, the distance of charge 3 from the origin because different distances give different values for the net force. You set that expression equal to the force on charge 3 , which is given to you, and then solve for the particular value of y that matches the given force. You cannot do this problem without doing algebra.

 

[boxybrownmd]

If you find an expression for the net force on charge 3, it will be a function of y, the distance of charge 3 from the origin because different distances give different values for the net force. You set that expression equal to the force on charge 3 , which is given to you, and then solve for the particular value of y that matches the given force. You cannot do this problem without doing algebra.

Wouldn't the expression for net force just be SQRT((Fx)^2 + (Fy)^2)? Fx would become 0, Fy would be 3.45x10^-5?

 

[kuruman]

I do not understand where the 3.45x10^-5 comes from. It is irrelevant. The expression for the magnitude of the net force is indeed $F_{net} = \sqrt {F_x ^2+F_y ^2}$. It is also true that $F_x = 0$, but it is not true that $F_y = 3.45 \times 10^5$. $F_y$ is the sum of two terms, that is the y-component of the force due to charge 1 added to the y-component of the force due to charge 2. You cannot set anything equal to 6.9x10^-5N unless you find an algebraic expression for the net force that has in it the distance of charge 3 (what you are looking for) from the origin.

I am starting to believe that your resistance to what I asked you to do as a starting point, namely to draw the forces acting on charge 3, may be arising from a lack of understanding of what I am really asking you to do or from your inability to do what I am asking because you don't know how to draw vectors and determine their components. Look at post #13. Is there anything that you do not understand or do not know how to do? If so, ask me and I will try to clarify it. I gave you a starting point and a road map to the solution (post #15). Follow it even though you may not be able to see the solution from where you are right now.

 

[boxybrownmd]

I do not understand where the 3.45x10^-5 comes from. It is irrelevant. The expression for the magnitude of the net force is indeed $F_{net} = \sqrt {F_x ^2+F_y ^2}$. It is also true that $F_x = 0$, but it is not true that $F_y = 3.45 \times 10^5$. $F_y$ is the sum of two terms, that is the y-component of the force due to charge 1 added to the y-component of the force due to charge 2. You cannot set anything equal to 6.9x10^-5N unless you find an algebraic expression for the net force that has in it the distance of charge 3 (what you are looking for) from the origin.

I am starting to believe that your resistance to what I asked you to do as a starting point, namely to draw the forces acting on charge 3, may be arising from a lack of understanding of what I am really asking you to do or from your inability to do what I am asking because you don't know how to draw vectors and determine their components. Look at post #13. Is there anything that you do not understand or do not know how to do? If so, ask me and I will try to clarify it. I gave you a starting point and a road map to the solution (post #15). Follow it even though you may not be able to see the solution from where you are right now.

I do not understand what you are trying to say in post #15. The reason why I said Fy = 3.45 x 10^-5 is because the problem states that particle 3 experiences a force from particle 1 and 2 equal to 6.9 × 10^−5 N ȷ^. I assumed that j^ represented force acting in the y direction, and got 3.45 x 10^-5 by taking half of the original force acting in the y direction so that I could find the amount of force a single particle exerts on particle 3 and figure out the distance from that point.

 

[kuruman]

OK, look at the figure that I attached. It is what I asked you to do as a starting point. O marks the origin of the axes and x₁, x₂ are the positions of charges q₁, q₂ on the x-axis. Charge 3 is at distance y on the y-axis. Study the drawing. F₁ and F₂ are the forces from each charge. Can you find an expression for the net force, i.e. the vector sum of F₁ and F₂? Clearly, your expression must involve y because different values of y give different values for the net force.

 

[boxybrownmd]

OK, look at the figure that I attached. It is what I asked you to do as a starting point. O marks the origin of the axes and x₁, x₂ are the positions of charges q₁, q₂ on the x-axis. Charge 3 is at distance y on the y-axis. Study the drawing. F₁ and F₂ are the forces from each charge. Can you find an expression for the net force, i.e. the vector sum of F₁ and F₂? Clearly, your expression must involve y because different values of y give different values for the net force.
View attachment 105801

Ok, so I had something somewhat similar drawn when I first attempted the problem a few days ago. However, breaking it down is giving me a few issues. F1 and F2 represent the total force of particle 1 and particle 2. When you take both of the x components of F1 and F2, they will cancel out because they are equal but opposite. However, all that would be left would be the sum of the y force, which would be the 6.9 x 10^-5, correct? Which means that wouldn't just be the y component of the force, but also the net force acting on particle 3? As for an expression, the only one I can think of is just using the vector sum equation in this case, but considering the x components cancel, there isn't much to it.

 

[kuruman]

F₁ has a y-component which is equal to half of 6.9x10^-5N only if q₃ is placed at the correct position on the y-axis. How can you find that position? You need to write a general expression for F_1y that is true for any value of F_net and then substitute the value of F_net that is given to you in that expression to find the particular value of y that fits the bill. You cannot find an unknown quantity without an equation. The 6.9x10^-5N is only the right side of the equation. What is the left side?

 

[boxybrownmd]

F₁ has a y-component which is equal to half of 6.9x10^-5N only if q₃ is placed at the correct position on the y-axis. How can you find that position? You need to write a general expression for F_1y that is true for any value of F_net and then substitute the value of F_net that is given to you in that expression to find the particular value of y that fits the bill. You cannot find an unknown quantity without an equation. The 6.9x10^-5N is only the right side of the equation. What is the left side?

Ok, I understand what you mean now. So basically, rather than find an equation that just finds distance for 6.9x10^-5, I need to find one that works for any net force to find a specific value of y? The only issue now is trying to figure out how to find an equation that would work from what I have. The only thing I can think of is using:

F1y = SQRT((F1)^2 - (F1x)^2)

When you say substitute the value of Fnet given to me in the expression, where am I looking exactly?

Sorry if I seem rather naive to this, I have little experience in force vectors in terms of problem solving and hoping to learn from this.

 

[kuruman]

Have you ever resolved a two-dimensional vector into components? If not, you need to learn how to do that before you tackle this problem.

 

[boxybrownmd]

Have you ever resolved a two-dimensional vector into components? If not, you need to learn how to do that before you tackle this problem.

Yes I have, but this just seems more complex with two different forces.

 

[kuruman]

It just seems that way, but it is not. Look at the drawing. Can you use Coulomb's law to find an expression for the magnitude of F₁ and write it down for me to see?

 

[boxybrownmd]

https://scontent.fewr1-1.fna.fbcdn.net/v/t34.0-12/14264093_1254405171250525_3984730770686346342_n.jpg?oh=346e65c981cd752be738e3496319668d&oe=57D9DCD3

This would be the magnitude of just F1, correct?

 

[kuruman]

You have misunderstood Coulomb's law. It states that the magnitude of the force between two charges is equal to constant (1/4πε₀) times the product of the charges divided by the square of the distance between the two charges. Look at the drawing. What is the square of the distance between q₁ and q₃? Do NOT use numbers; use the symbols that are shown in the drawing.

 

[boxybrownmd]

You have misunderstood Coulomb's law. It states that the magnitude of the force between two charges is equal to constant (1/4πε₀) times the product of the charges divided by the square of the distance between the two charges. Look at the drawing. What is the square of the distance between q₁ and q₃? Do NOT use numbers; use the symbols that are shown in the drawing.

I sincerely do not know. I thought the distance between q1 and q3 was just SQRT((x^2) + (y^2)).

 

[kuruman]

It is, but I asked you to find the square of that distance. You need the square because it is what appears in the denominator when you write Coulomb's law. So find the square of the distance and divide it into (1/4πε₀) times the product of the charges to find F₁.

I have to teach my class now, be back in an hour.

 

[boxybrownmd]

It is, but I asked you to find the square of that distance. You need the square because it is what appears in the denominator when you write Coulomb's law. So find the square of the distance and divide it into (1/4πε₀) times the product of the charges to find F₁.

I have to teach my class now, be back in an hour.

Here is what I have so far. Is this the right process?
https://scontent.fewr1-1.fna.fbcdn.net/v/t34.0-12/14344135_1254552687902440_1826194822450464519_n.jpg?oh=dc4822bd9ff309ed6c57b7d5d10a611b&oe=57D97E88

 

[kuruman]

It's fine, but you really need to learn to use symbols instead of numbers. It's much easier, shorter and neater to write k instead of 9x10⁹, q₁ instead of 6x10^-9, q₃ instead of 2x10^-9 and x instead of 0.03. The main reason for this is that it will be much easier to troubleshoot your algebra if something goes wrong. Anyway, now that you have the magnitude of force F₁, can you write expressions for its x and y components?

 

[boxybrownmd]

It's fine, but you really need to learn to use symbols instead of numbers. It's much easier, shorter and neater to write k instead of 9x10⁹, q₁ instead of 6x10^-9, q₃ instead of 2x10^-9 and x instead of 0.03. The main reason for this is that it will be much easier to troubleshoot your algebra if something goes wrong. Anyway, now that you have the magnitude of force F₁, can you write expressions for its x and y components?

To write expressions for the x and y components, would I just move the y to the left hand side and get the expression for just y, and then do the same for x?

 

[kuruman]

I am not sure I understand what you are asking, but you need to end up with a right triangle in which F₁ is the hypotenuse, F_1x is the horizontal right side and F_1y is the vertical right side. Perhaps you can show me a picture before you proceed further.

 

[haruspex]

To write expressions for the x and y components, would I just move the y to the left hand side and get the expression for just y, and then do the same for x?

You might find it easier if you put in a variable for the angle the electrostatic force makes to the x axis, θ say.
In terms of that, what is the component of the force along the y axis?