Three charges constrained to a ring

In summary, the three charges are constrained to a ring of radius r. Charges 1 and 3 are constrained to have angles ϑ12 and ϑ13 with each other. Charges 2 and 3 are constrained to have angles ϑ12 and ϑ13 with charge 1. 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.
  • #1
auctor
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Homework Statement



Three non-equal positive point charges, q1, q2, and q3, 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?
 
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  • #2
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.
 
  • #3
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?
 
  • #4
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).
 
  • #5
Ok, thanks! I guess I was overthinking this. Will do the second force balance.
 

FAQ: Three charges constrained to a ring

1. What is the concept behind "Three charges constrained to a ring"?

The concept involves three point charges that are placed on a ring and are constrained to move only along the ring's circumference. The charges experience forces due to their interactions with each other and the ring, resulting in a stable equilibrium position.

2. How are the forces between the charges calculated in this scenario?

The forces between the charges can be calculated using Coulomb's law, which states that the force between two point charges is directly proportional to the product of their charges and inversely proportional to the square of the distance between them.

3. What is the significance of the ring in this setup?

The ring acts as a constraint for the charges, ensuring that they remain at a fixed distance from each other and preventing them from moving in a linear path. It also provides a stable equilibrium position for the charges.

4. Can the charges be of any magnitude and still remain in equilibrium on the ring?

No, the charges must be of equal magnitude in order to remain in equilibrium on the ring. If one charge is larger than the others, it will experience a stronger force and will move out of the equilibrium position.

5. How does the distance between the charges affect the equilibrium position?

The distance between the charges affects the equilibrium position by altering the forces between them. As the distance increases, the forces decrease, causing the charges to move further away from each other and away from the equilibrium position. Conversely, decreasing the distance between the charges increases the forces and brings them closer to the equilibrium position.

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