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Homework Statement
Figure 2134 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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