Charge distribution along a square loop in equilibrium

[NEUR]

Homework Statement

A conductor (wire) is folded into a square loop with each side having a length of a. Total charge of Q is transferred onto the conductor. Describe the line charge density of the square loop in equilibrium. (If I am interpreting this correctly what is required is determining charge distribution along the wire.)

Homework Equations

Guessing here:
E=(1/4πε₀)(dQ/r²) - electric field of a point charge

The Attempt at a Solution

If charge distribution would be homogenus all of this would be easy. q=Q/(4a) But I take it this is not the case, correct?

Perhaps taking into account that electric field component parallel to the wire at each point on the square equals zero since everything is in equilibrium would help? But I can not find a way to set any equations.

Sum of electric fields caused by infinitesimal charges along the wire in the midle of the square equals zero. But again, I see no way of using that to set an equation which would show charge distribution along the wire.

Since we are dealing with a square, symmetry probably helps out a lot and a solution for only a half of one side would solve everything. Correct?

All help is greatly appreciated.

Cheers!

 

[voko]

There may be a simpler way, but one approach that immediately springs to mind is to find the distribution that minimizes potential energy. And you are quite right about the symmetry.

 

[tiny-tim]

Hi NEUR! Welcome to PF!

Perhaps taking into account that electric field component parallel to the wire at each point on the square equals zero since everything is in equilibrium would help? But I can not find a way to set any equations.

Call the charge density f(x) at distance x from the centre of any side (x ≤ a/2).

Then either find the equation for the component of the field to be zero (as you suggested),

or find the equation for the potential to be constant.