Equilibrium of Charges at Circumference of a Circle

[Buffu]

Homework Statement

Two charges placed at circumference of a circle of radius $a$ at $\pi/2$ from each other. Find the relative magnitude of third charge kept on the circumference such that the system is at equilibrium.

Homework Equations

Coulombs law.

The Attempt at a Solution

Let $Q$ be the unknown charge and $x$ be the length of equal sides of the triangle . then I get,

By coulombs law,

$$F_{BA} = {-Qq \over x^2 } \left(\cos(135/2)\hat i + \hat j \sin(135/2) \right)$$

$$F_{CA} = {-q^2\over 4x^2(\cos (135/2))}(\hat i) $$Now the force on $A$ should be zero for the system to be in equilibrium but clearly there is a net force in -y direction that is not balanced by anything, so how the system is in equilibrium for any value of $Q$ ?

Where did I go wrong ?

 

[kuruman]

What is the value of $x$?

 

[Buffu]

What is the value of $x$?

Numerical value or what it represent in equations ?

 

[kuruman]

Numerical value or what it represent in equations ?

It's OK, I figured it out from the way you used it. I think you misunderstood that the problem requires the charges to be constrained on the circle. Imagine three charged beads on a wire loop, for example. As you correctly discovered, the force on anyone bead cannot be zero. However, if you constrained the charges on the (non-deformable) circle, equilibrium is reached if the net force on anyone bead is directed radially out. Therefore, the task before you is to balance the tangential forces.

 

[kuruman]

Upon doing the problem, I think you will benefit from expressing the two charge-to-charge distances in terms of the radius of the circle. To do that, draw the inner right triangle (with its apex at the center) and then use the law of sines.

 

[Buffu]

It's OK, I figured it out from the way you used it. I think you misunderstood that the problem requires the charges to be constrained on the circle. Imagine three charged beads on a wire loop, for example. As you correctly discovered, the force on anyone bead cannot be zero. However, if you constrained the charges on the (non-deformable) circle, equilibrium is reached if the net force on anyone bead is directed radially out. Therefore, the task before you is to balance the tangential forces.

Thank you, I will try and tell you the results.