- #1

AndersF

- 27

- 4

- 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?

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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