# Potential energy of a system of two punctual charges along the X axis

• AndersF
In summary, the conversation was about solving a problem involving the work needed to move a third charge from infinity to the origin in a system of two punctual charges. The conversation included discussions about the potential energy and distances between the charges, as well as alternative approaches to finding the solution. The key takeaway was that the correct expression for potential energy should include absolute value signs to account for negative distances, leading to the correct result according to the textbook.
AndersF
Homework Statement
Two punctual charges ##+Q## are placed along the ##X##-axis at the points ##x=−a## and ##x=a##. Find the work to move a third charge ##−Q## from the infinity to the origin.
Relevant Equations
##E_{p}=-k \frac{Q^2}{r}##
I have not clear how to solve this problem. Here it is my attempt at a solution:

Let the charge at ##-a## be the number one and the one at ##+a## the number two. the potential energy of the punctual charge ##-Q## due to each charge +Q will be then ##E_{pi}=-k \frac{Q^2}{r_i}##, whit ##r_i## the distance between then. So, if the charge ##-Q## is at a distance ##x## from the origin, the total potential energy will be the sum of those ones due to each charge ##+Q##:

##E_{p_1}=-k \frac{Q^2}{x+a}##
##E_{p_2}=-k \frac{Q^2}{x-a}##

##E_{p}=E_{p_1}+E_{p_2}=-kQ^2 (\frac{1}{x+a}+\frac{1}{x-a})##

As the electric field is conservative, the work from ##\infty## to ##0## will be the difference of potential energy between those points, so:

##W_{-\infty,0}=\Delta E_p=E(0)-E(-\infty)=0##

But this is not the solution at my textbook... Am I missing something?

By the way: as the energy needed to cross the ##x=-a## point is infinite, how would it be possible to arrive at the ##O## point?

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I don't recall having heard this term before: "punctual charge".
As near as I can tell it refers to a point charge.

AndersF said:
By the way: as the energy needed to cross the ##x=-a## point is infinite, how would it be possible to arrive at the ##O## point?

You don't have to assemble the charges in this way. The electric force is conservative, which means that the work it does only depends on the start and end points. You could start off with the third charge at infinity, bring it toward the others, do a few laps around the other two, use it to play a quick game of table tennis and then finally slot it into position. The work done by the electric field on this charge will be the same!

AndersF said:
Homework Statement:: Two punctual charges ##+Q## are placed along the ##X##-axis at the points ##x=−a## and ##x=a##. Find the work to move a third charge ##−Q## from the infinity to the origin.
Relevant Equations:: ##E_{p}=-k \frac{Q^2}{r}##

I have not clear how to solve this problem. Here it is my attempt at a solution:

Let the charge at ##-a## be the number one and the one at ##+a## the number two. the potential energy of the punctual charge ##-Q## due to each charge +Q will be then ##E_{pi}=-k \frac{Q^2}{r_i}##, whit ##r_i## the distance between then. So, if the charge ##-Q## is at a distance ##x## from the origin, the total potential energy will be the sum of those ones due to each charge ##+Q##:

##E_{p_1}=-k \frac{Q^2}{x+a}##
##E_{p_2}=-k \frac{Q^2}{x-a}##

##E_{p}=E_{p_1}+E_{p_2}=-kQ^2 (\frac{1}{x+a}+\frac{1}{x-a})##

As the electric field is conservative, the work from ##\infty## to ##0## will be the difference of potential energy between those points, so:

##W_{-\infty,0}=\Delta E_p=E(0)-E(-\infty)=0##

But this is not the solution at my textbook... Am I missing something?

By the way: as the energy needed to cross the ##x=-a## point is infinite, how would it be possible to arrive at the ##O## point?
When the location of the third charge is such that ## -a < x < a ##, your distances are incorrect. Distance is a non-negative quantity.

Another idea is to approach origin along the y-axis.

AndersF
SammyS said:
I don't recall having heard this term before: "punctual charge".
As near as I can tell it refers to a point charge.
It's true, I was referring to point charges xD

SammyS said:
When the location of the third charge is such that ## -a < x < a ##, your distances are incorrect. Distance is a non-negative quantity.

Another idea is to approach origin along the y-axis.
Ok, so the expression of the potential energy should then be written in a different manner to avoid negative distances? What would be the equation?

I hadn't written it, but the correct result, according to my textbook, should be ##W=\Delta E_{p}=-2 k \frac{Q^{2}}{a}##

SammyS said:
When the location of the third charge is such that ## -a < x < a ##, your distances are incorrect. Distance is a non-negative quantity.

Another idea is to approach origin along the y-axis.

Ok, thanks, here it is the correct expression:

##E_{p}=E_{p_1}+E_{p_2}=-kQ^2 (\frac{1}{|x+a|}+\frac{1}{|x-a|})##

AndersF said:
Ok, thanks, here it is the correct expression:

##E_{p}=E_{p_1}+E_{p_2}=-kQ^2 (\frac{1}{|x+a|}+\frac{1}{|x-a|})##
Correct. Using this expression should give you the correct result as given by the textbook.

By the way:

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## 1. What is potential energy in a system of two punctual charges along the X axis?

Potential energy is the energy that a system of two punctual charges possesses due to their positions and interactions with each other. It is a measure of the work that would be required to bring the two charges from an infinite distance apart to their current positions.

## 2. How is potential energy calculated in this system?

The potential energy of a system of two punctual charges along the X axis can be calculated using the formula U = k(q1*q2)/r, where k is the Coulomb's constant, q1 and q2 are the charges of the two particles, and r is the distance between them.

## 3. What is the relationship between potential energy and distance in this system?

In this system, potential energy is inversely proportional to the distance between the two charges. As the distance between the charges increases, the potential energy decreases, and vice versa.

## 4. How does the sign of the charges affect the potential energy in this system?

The sign of the charges plays a crucial role in determining the potential energy in this system. If the charges are of the same sign, the potential energy is positive, indicating a repulsive force between the charges. If the charges are of opposite signs, the potential energy is negative, indicating an attractive force between the charges.

## 5. What is the significance of potential energy in this system?

Potential energy is an essential concept in understanding the behavior of charged particles in a system. It helps in predicting the forces between the particles and their movements. It is also a fundamental concept in the study of electromagnetism and plays a crucial role in many practical applications, such as in the design of electronic devices and circuits.

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