How Is Energy Calculated for Four Charges at a Square's Corners?

[Riogho]

What is the energy required to bring four identical charges in from infinity such that they occupy the corners of a square with side length r?

 

[dynamicsolo]

I'm presuming you're presently learning about electrostatic potential energy. You approach the problem by imagining assembling the square charge by charge in "empty space". The first charge can come in for free, since there's nothing else around. The second charge comes in from infinity down to a distance r from the first charge. Recall the relationship between work done on that second charge and the value of the final potential energy between the two charges.

You then continue this by placing the third charge at one of the remaining corners of the square, then finally placing the fourth charge at the vacant vertex. Each additional does not affect the results from the previous stages; the total work done will just be the sum for each of these three steps.

 

[Moetasim]

The energy required to bring four identical charges in from infinity and arrange them at the corners of a square with side length r can be calculated using the Coulomb's Law. This law states that the force between two point charges is directly proportional to the product of their charges and inversely proportional to the square of the distance between them.

In this problem, we have four identical charges, so the total energy required would be four times the energy required to bring one charge in. The distance between the charges would be the diagonal of the square, which can be calculated using the Pythagorean theorem as r√2.

Using Coulomb's Law, the energy required to bring one charge in from infinity and place it at one of the corners of the square would be given by:

E = k(q^2/r√2)

Where k is the Coulomb's constant and q is the charge of each particle.

Therefore, the total energy required to bring four identical charges in from infinity and arrange them at the corners of a square with side length r would be:

E_total = 4k(q^2/r√2)

This equation shows that the energy required is directly proportional to the charge of each particle and inversely proportional to the square root of the distance between the charges. As the distance between the charges increases, the energy required decreases, and as the charge of the particles increases, the energy required also increases.

In conclusion, the energy required to bring four identical charges from infinity and arrange them at the corners of a square with side length r can be calculated using Coulomb's Law and is directly proportional to the charge and inversely proportional to the square root of the distance between the charges.