Three charges constrained to a ring

[auctor]

Homework Statement

Three non-equal positive point charges, q₁, q₂, and q_3, are constrained to a ring of radius r. Find their relative positions in equilibrium using force balances.

Homework Equations

I know that this problem could be approached in two ways: using force balances or minimizing the total energy. The statement specifically asks to use force balances. The only relevant equation, then, is Coulomb's law:

F=q₁*q₂/(4*π*ε₀R²).

The Attempt at a Solution

Define angles between charges 1 and 2 and between charges 1 and 3: ϑ₁₂ and ϑ₁₃. These completely define the relative positions of all 3 charges.

Express the distances between charges 1-2 and 1-3 in terms of these angles and the radius: r₁₂=2*r*sin(ϑ₁₂/2) and r₁₃=2*r*sin(ϑ₁₃/2), from simple geometrical considerations. The forces can then be expressed in terms of the radius and the angles using Coulomb's law.

The total force on charge 1 is the vector sum of the Coulomb forces of its interactions with charges 2 and 3, both repulsive and a force that constrains the charge to the ring, which must be directed towards the center of the ring. Therefore, the sum of components of the two Coulomb forces that are tangential to the ring (perpendicular to its radius) should be zero.

Again, from simple geometrical and trig considerations the magnitudes of these components can be found to be F₁₂*cos(ϑ₁₂/2) and F₁₃*cos(ϑ₁₃). Equating these magnitudes and cancelling out whatever is possible gives

q₂/q₃ = [sin(ϑ₁₂/2)*tan(ϑ₁₂/2)]/[sin(ϑ₁₃/2)*tan(ϑ₁₃/2].

This relates the angles with the ratio of charges 2 and 3. However, charge 1 doesn't seem to enter the picture, which seems wrong. (I am guessing, the answer should be independent of the ring's radius.) Where am I going wrong?

 

[mfb]

You just considered the forces on charge 1 here - in equilibrium it won't depend on charge 1 (twice the charge just gives you twice the force, but 2*0=0).
q1 is important for the force balances of the other two particles.

 

[auctor]

I see, but doesn't the answer I got already define the relationship between ϑ₁₂ and ϑ₁₃? I.e., wouldn't considering the force balance for another charge create more constraints than there are degrees of freedom?

 

[mfb]

You got a relationship between the two, but you don't have values for the two yet.
Adding the second force balance will give you values for the two angles.
The third force balance is redundant, it follows from the first two force balances (no matter which one you choose as third one).

 

[auctor]

Ok, thanks! I guess I was overthinking this. Will do the second force balance.