# Finding θ That Balances the Electrical Forces

## Homework Statement

Figure 21-34 shows electrons 1 and 2 on an x axis
and charged ions 3 and 4 of identical
charge q and at identical
angles θ. Electron 2 is free to
move; the other three particles
are fixed in place at horizontal
distances R from electron 2 and
are intended to hold electron 2 in
place. For physically possible values
of q $\leq$ 5e, what are the (a)
smallest, (b) second smallest, and
(c) third smallest values of θ for
which electron 2 is held in place?

## Homework Equations

Fx,2,net= F21 - cosθ(F24+F23)

2y = The distance from q3 and q4

I created a FBD for q2

## The Attempt at a Solution

Okay, so I derived the above equation from the sum of the forces in the x direction. I'm trying to solve for cosθ, and the equation I ended up with is:

cosθ = (-e/2q3)1/3

I know that θ≥0 because q3 and q4 can't occupy the same space.

I also know that in order for F2,net to equal zero, F21 must equal cosθ(F24+F23).

Therefore: kq1q2/R2 = cosθ[(kq2q4)+(kq2q3)/(R2+y2)]

..and that's where I get lost. I end up with the cosθ equation from above. I feel like I should be taking a derivative. Perhaps δθ/δq3. Could someone please point me in the right direction. I tried working on it for quite some time but now my ADHD meds are wearing off and I'm about to take a nosedive directly into keyboard land. Thank you for any help.

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ehild
Homework Helper
You have an equation for cos(theta). Think what values q can have. Can an ion be charged with a fraction of the elementary charge e? Can be q positive?

ehild

Okay, I understand that, so what i did was find out what θ would be if q3 were certain integer values under 5e. I set up a table, and now I'm trying to figure out what F2,net is going to be with that value of q3 and θ. But I've been ending up with the answers in terms of R2. How would I go about solving for that? Thanks for your help by the way.

ehild
Homework Helper
The problem asks the possible values of theta. You get it from the equation cosθ = (-e/2q3)1/3. Why do you want to find the value of the force?

ehild

Because it's asking for what values of θ is q2 held in place. So F2,net would need to be 0 at those particular θ's

Okay, I might have gotten my answers. What I did was to solve for F2,net, plugging in the values of q3 and θ into it. But what I found is that for each discrete value of q3, the net force on q2 was 0.

for example, if

q3 = e

→ θ ≈ 37.467o

and solving for F2,net

F2,net = 0 = q1/R2 - cos3θ(2q3/R2) = -e + 1/2(2e)/R2 = 0

(Notice that I factored out coulombs constant and q2. Also notice that R2 = cos2θ(R2 + y2)

Now I could just use the lowest three values of θ for my answer, but I have an underlying suspicion that I did something wrong. Could someone please tell me if I need to rework it?
What's basically happening is that cos3θ is cancelling out the value of q3, which is making every solution (-e + e)/R2, which is obviously 0.

ehild
Homework Helper
You got the equation cosθ = (-e/2q3)1/3 from the condition that F2,net=0. Why is it surprising that F2net really zero? q3 and q4 are identical charges and they must be negative otherwise the total force on electron2 can not be zero.

q3=q4=q =-n e where n is integer and less than 5. It can be 1,2,3,4. Calculate theta for all.

ehild

I understand now! Thank you, I just wasn't connecting the fact that the cosθ equation is based on F2,net being 0, thanks. You told me exactly what I needed to hear.