Charge distribution along a square loop in equilibrium

In summary, the charge on the wire is distributed according to an equation that minimizes potential energy. However, this equation is not directly applicable because of symmetry.
  • #1
NEUR
2
0

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!
 
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  • #2
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.
 
  • #3
Hi NEUR! Welcome to PF! :smile:
NEUR said:
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. :wink:
 

Related to Charge distribution along a square loop in equilibrium

1. What is the concept of charge distribution along a square loop in equilibrium?

The concept of charge distribution along a square loop in equilibrium refers to the distribution of electric charges on a square-shaped loop that is not moving or accelerating. This is a common scenario in electrostatics, where the charges are at rest and the net force on each charge is zero.

2. How is the charge distributed along a square loop in equilibrium?

In a square loop in equilibrium, the charge distribution is uniform. This means that the charges are spread out evenly along the loop, with the same amount of charge on each side. This is due to the repulsive forces between like charges, which causes them to spread out as much as possible.

3. What factors affect the charge distribution along a square loop in equilibrium?

The charge distribution along a square loop in equilibrium is affected by the charge density, the size of the loop, and the distance between the charges. As the charge density or the size of the loop increases, the charges will be more spread out along the loop. Similarly, as the distance between the charges increases, the charges will be less spread out.

4. How is the electric field affected by the charge distribution along a square loop in equilibrium?

The electric field around a square loop in equilibrium is affected by the charge distribution. The electric field lines will be perpendicular to the loop and will be stronger near the corners of the loop where the charges are closer together. The electric field will be weaker along the sides of the loop where the charges are more spread out.

5. What is the significance of studying charge distribution along a square loop in equilibrium?

Studying charge distribution along a square loop in equilibrium is important in understanding the behavior of electric charges in static situations. It also helps in understanding the concept of electrostatic equilibrium and the relationship between electric fields and charge distribution. This knowledge is essential in many fields, including electronics, physics, and engineering.

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