Two Charged Masses Suspended From Strings

[Geromy]

Homework Statement

Two similar tiny balls of mass m are hung from silk threads of length L and carry equal charges q. The angle formed by the two strings is bisected by an imaginary line, forming angle θ. Assume that θ is so small that tan θ can be replaced by its approximate equal, sin θ. The distance between the two charged masses, at equilibrium, is x. Show that, for equilibrium,

x = ((q²L)/(2∏ε₀mg))^1/3

Homework Equations

Clearly, Coulomb's Law is a vital piece of this problem, and other sources that I've checked suggest such things as the constancy of the ratio of the sides of a triangle divided by the sine of that side's opposite angle and combining the electric force and gravitational force into the components of the force of tension.

The Attempt at a Solution

Despite having a couple of ideas about how to start, I haven't made much progress. I noticed that there is an identical problem here (https://www.physicsforums.com/showthread.php?t=305517) that suggested the constant sine ratio rule, but I can't for the life of me get to the right answer. Any help would be appreciated.

 

[Geromy]

Haha, so, I figured it out! I thought I would post here in case anyone wants an explanation.

Consider the forces acting on a single one of the charged masses: the force due to the charges is kq²/x² - x is the radius between the two charges, since it's the distance they're separated by.

The force due to gravity is also pretty straightforward: it's simply mg.

Since these two forces are perpendicular, they form two sides of a right triangle. The hypotenuse of this triangle is equal to the final force acting on the charged mass, the force of tension. This triangle contains angle θ, since the force of tension is in the direction of the silk thread.

Since now we've reintroduced θ, we can start using trigonometry. Tan θ = Opp/Adj, which in this case is our force due to electric charge (F) over the force due to gravity (W), or F/W.

However, Tan θ = Sin θ, which equals x/2L

when you combine equations, you get 2kq²L = mgx³, and from here it's a simple matter of algebra to get to the final equation.

 

[ehild]

Well done!

Welcome to PF!

ehild