Finding θ That Balances the Electrical Forces

[Legaldose]

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

F_x,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 F_2,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.

 

[ehild]

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

 

[Legaldose]

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 F_2,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]

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

 

[Legaldose]

Because it's asking for what values of θ is q₂ held in place. So F_2,net would need to be 0 at those particular θ's

 

[Legaldose]

Okay, I might have gotten my answers. What I did was to solve for F_2,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 ≈ 37.467_o

and solving for F_2,net

F_2,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]

You got the equation cosθ = (-e/2q₃)^1/3 from the condition that F_2,net=0. Why is it surprising that F_2net 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

 

[Legaldose]

I understand now! Thank you, I just wasn't connecting the fact that the cosθ equation is based on F_2,net being 0, thanks. You told me exactly what I needed to hear.