Net Electric Force on +q Charge with Two Added Charges

[aeromat]

Homework Statement

Two more +Q charges are held in place the same distance "s" away from the +q charge as shown. Consider the following student dialogue concerning the net force on the +q charge:

The two added charges are written with blue Q's

Student 1: "The net electric force on the +q charge is now three times as large as before, since there are now three positive charges exerting forces on it."

Student 2: "I don't think so. The force from the +Q charge on the left will cencel the force from +Q charge on the right. The net electric force will be the same as in part A."

Homework Equations

Coulomb's Law

The Attempt at a Solution

What I did was I drew out that diagram above on a piece of paper and each line I drew pointing from the +q to the Q was exactly the same length. Next, I drew the force vectors after(in this case, it would be repulsion), and they ended up looking something like this:

I found that it was less than three times one of the forces. Also, they could'nt have canceled each other out because they were repulsive. However, I know I could show this through visually, but how could I prove it mathematically (using physics formulae) that this is correct?

 

[eczeno]

don't discount the value of a proof by drawing. you have shown that the total force is more than before, not not 3 times more, so both students are wrong.

if you do want to show it with math, you need to calculate the force (remember it is a vector) from each charge and then add them together (using vector addition) to get the total force. it is exactly what you have done in your drawing, but with symbols. if you want, i can show you how that is done, but i like your proof just fine.

one thing you say is wrong. the two new charges don't cancel each other out completely, but they do a little, that is why there is not 3 times as much force. if they were placed in such a way that q was half way between them and on the line connecting them, then they would cancel completely. it would be like two people pushing you, one form each side; you won't go anywhere.

hope this helps

 

[aeromat]

if you want, i can show you how that is done

Yes, please if you can. I understand what you mean by them not cancelling out completely, thank you for your help on that.

 

[eczeno]

ok, i won't go through all the gory details of using coulomb's law, i will just go through adding the vectors. we can fill in the details later if you want.

so, we have three vectors, call them A, B and C, and these will be the forces from the charges starting from the left. so, in your figure, A is the bottom vector, B is the vertical one, and C is the top one.

hmm... let me get a better feel for what you know of vectors before i go on. a few questions for you:

have you ever worked with vectors before?

do you know how to find the components of a vector?

 

[aeromat]

ok, i won't go through all the gory details of using coulomb's law, i will just go through adding the vectors. we can fill in the details later if you want.

so, we have three vectors, call them A, B and C, and these will be the forces from the charges starting from the left. so, in your figure, A is the bottom vector, B is the vertical one, and C is the top one.

hmm... let me get a better feel for what you know of vectors before i go on. a few questions for you:

have you ever worked with vectors before?

do you know how to find the components of a vector?

Alright this is what I know about vectors:
> Dot/Cross Product
> Splitting a vector into its x/y components
> Scalar/Vector projections

So yes, I do have a good base when it comes with vectors.

 

[eczeno]

oh, great, then this will be easy. Each force has the same magnitude F, but the blue forces are not exactly vertical, so they have components Fx and Fy, but, while they have the same y component, they x components will be in opposite directions.

so, when we add the forces together, we add all the x and all the y components, like this:

F_{tot}_x=F_x-F_x=0

and

F_{tot}_y=F+F_y+F_y<3F

 

[aeromat]

So if the Fx forces did not cancel each other out, then the net force would be 3 times as great?

 

[eczeno]

maybe; say you put all three charges in a clump where the original Q was, then the force would be 3x greater. but, if you put both charges together on one side, but away from the bottom charge. then their x components would add, and so would their y, but the total force would still be < 3x bigger. this is because the bottom charge and the two blue charges would be, in a sense, fighting against each other a little bit.