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I would like to thank all of the contributors on this site. You have helped me in more ways than I can count. I am struggling with the following concept and was wondering if anyone could clarify this.

As the electron gets further away from the nucleus, its energy increases. This makes sense when we look at the work equation, we do more work to pull it further away from where it wants to be.

What I am struggling with is the coulomb equation, where the the force is inversely proportional to distance squared. As the distance increases the force should decrease. Then, if we were to integrate that force to get work, the work would be inversely proportional to the distance, which tells us that work or energy should decrease as you get further away. I am not sure where my mistake is. Any help is much appreciated