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I have a source charge distribution $A^\prime$. I have another charge distribution $A$ of the same sign. Will the potential energy in moving charge distribution $A$ towards source charge distribution $A^\prime$ be always positive irrespective of whether the charges are positive or negative?

My try:

Potential energy in moving a point charge towards another source point charge (of the same sign) is:

$$\phi=\dfrac{kqq'}{r}-\dfrac{kqq'}{r_0}$$ Since we are moving towards source point charge, $r_0>r$ and hence potential energy $(\phi)$ is posiitve.

$$\text{OR}$$

$$\phi=\dfrac{k(-q)(-q')}{r}-\dfrac{k(-q)(-q')}{r_0}=\dfrac{kqq'}{r}-\dfrac{kqq'}{r_0}$$ which is again positive for the same reason.

By applying the same reasoning and adding up the infinitesimal charges, I think the answer to my question is yes. Anything wrong in my reasoning?

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  • $\begingroup$ Why is everyone so silent on this question??? $\endgroup$
    – lorilori
    Jun 22, 2017 at 16:05
  • $\begingroup$ Are the charge distributions just two point-like particles? $\endgroup$
    – caverac
    Jun 22, 2017 at 16:12
  • $\begingroup$ Not necessarily. Could be of any shape. $\endgroup$
    – lorilori
    Jun 22, 2017 at 16:17
  • $\begingroup$ But couldn't we at first see how point charges behave and extend its property to charge distribution? $\endgroup$
    – lorilori
    Jun 22, 2017 at 16:18
  • $\begingroup$ If you take a finite charge distribution, it is true that at large distances the potential roughly behaves as $1/r$, but close to the distribution itself the story is a bit different, in particular the landscape of the potential could increase or decrease $\endgroup$
    – caverac
    Jun 22, 2017 at 16:29

1 Answer 1

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You are correct in saying that irrespective of the sign of the charge distributions( as long as they are both positive or both negative) if one charge is moving towards another charge there will be an increase in electric potential energy(as work needs to be done on a charge to overcome repulsive forces here as the force is repulsive between charges if they are both positive or negative).

The formulations involved you have already shown in your working.

Note: This will be true for as long as the charge distributions can each be considered point charges( like if they are two charged spheres, they shouldn't overlap for this to be true)

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  • $\begingroup$ Charge distributions are a collection of infinitesimally small point charges. What do you mean by "as long as the charge distributions can each be considered point charges"? $\endgroup$
    – lorilori
    Jun 22, 2017 at 16:46
  • $\begingroup$ In that case, their resultant is also a point charge hence what you have said is true. $\endgroup$ Jun 22, 2017 at 16:48
  • $\begingroup$ I mean the charge distribution has a finite volume. But a point charge is located at a point. If the charge distribution is of finite volume, then also will I be correct? $\endgroup$
    – lorilori
    Jun 22, 2017 at 16:53
  • $\begingroup$ Yes as long as the separate distributions don't overlap. $\endgroup$ Jun 22, 2017 at 16:54
  • $\begingroup$ I see. Do you mean while moving distribution $A$ towards distribution $A'$, the two distributions shouldn't overlap anywhere in the path of motion? $\endgroup$
    – lorilori
    Jun 22, 2017 at 17:06

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