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## 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 [itex]\leq[/itex] 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

F

_{x,2,net}= F

_{21}- cosθ(F

_{24}+F

_{23})

2y = The distance from q

_{3}and q

_{4}

I created a FBD for q

_{2}

## 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/2q

_{3})

^{1/3}

I know that θ≥0 because q

_{3}and q

_{4}can't occupy the same space.

I also know that in order for F

_{2,net}to equal zero, F

_{21}must equal cosθ(F

_{24}+F

_{23}).

Therefore: kq

_{1}q

_{2}/R

^{2}= cosθ[(kq

_{2}q

_{4})+(kq

_{2}q

_{3})/(R

^{2}+y

^{2})]

..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/δq

_{3}. 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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