Equilibrium arrangement angle of 4 charges in a square

[mik2334]

Homework Statement

2. This question starts of by presenting an arrangement of 4 equally positive charges attached by unstretchable strings, each of length l. The arrangement of the 4 charges in equilibrium can be described by the singular degree of freedom, θ, from the diagonal line between Q3 and Q2 to the horizontal.
• While we have not been given values for charges of for the angle θ, we can assume that this angle can vary depending on the ratio of charges, which will affect the overall shape of the arrangement (this implies a qualitative approach for θ, in which we can describe how it varies with varying charges)
• After each perturbation in charge, the angle must be calculated using two different methods: firstly, using vector methods (i.e. superposition of forces) and secondly using a scalar method (i.e. energies/Lagrangians of each charge)
• Each method should produce an identical result

Relevant Equations

F = kQ^2/r^2, U = kQ^2/r

My best attempt at this question is i have used Q3 as my origin, calculated all the horizontal and vertical forces acting on it due to the other charges, without inlcuding tension, and then I did tan(theta) = vertical forces/horizontal forces, which simplifies to an equation of than(theta) = Q1/Q4. I repeated this using Q1 for my origin, which also gave me the equation of tan(theta) = Q2/Q3. I do not know if this is correct, as i cannot show how the angle changes if Q2 increases and therefore the new arrangement of the charges. can i have some help pls

 

[BvU]

Just to clarify the problem statement:
With '4 equally positive' you mean '4 positive'
The values of the charges are unknown
The charges are coplanar
Right?

What is the condition for equilibrium for a given $\theta$ ? (I don't understand yours)

$\$

 

[Gavran]

I have found a similar problem where all four charges are positive and $Q_1=Q_4=Q$ and $Q_2=Q_3=q$.
The shape of the initial arrangement when $Q=q$ is a square and $\theta$ in that case will be $45^\circ$.
The shape of the arrangement when $Q \neq q$ is a rhombus and $\theta$ in that case will not be $45^\circ$.

@mik2334, can you check again your problem statement to make sure that all four charges are different in the general case.