# Is potential energy always positive in this case?

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?

• Why is everyone so silent on this question??? Jun 22, 2017 at 16:05
• Are the charge distributions just two point-like particles? Jun 22, 2017 at 16:12
• Not necessarily. Could be of any shape. Jun 22, 2017 at 16:17
• But couldn't we at first see how point charges behave and extend its property to charge distribution? Jun 22, 2017 at 16:18
• 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 Jun 22, 2017 at 16:29

• I see. Do you mean while moving distribution $A$ towards distribution $A'$, the two distributions shouldn't overlap anywhere in the path of motion? Jun 22, 2017 at 17:06