- #1

auctor

- 8

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## Homework Statement

Three non-equal positive point charges, q

_{1}, q

_{2}, 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=q

_{1}*q

_{2}/(4*π*ε

_{0}R

^{2}).

## The Attempt at a Solution

Define angles between charges 1 and 2 and between charges 1 and 3: ϑ

_{12}and ϑ

_{13}. 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: r

_{12}=2*r*sin(ϑ

_{12}/2) and r

_{13}=2*r*sin(ϑ

_{13}/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 F

_{12}*cos(ϑ

_{12}/2) and F

_{13}*cos(ϑ

_{13}). Equating these magnitudes and cancelling out whatever is possible gives

q

_{2}/q

_{3}= [sin(ϑ

_{12}/2)*tan(ϑ

_{12}/2)]/[sin(ϑ

_{13}/2)*tan(ϑ

_{13}/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?