[Physics] Relationship between work and energy in an electrostatic system

electrostaticspotential energywork

I am having problems understanding the relationship between the concepts of Work and Energy in a electrostatic system.

As I know, the definition of Potential Energy is the ability to do work. In a gravitational field, that would be $$\mathrm{PE} = W(\mathrm{after}) = -W(\mathrm{before})$$ where $W(\mathrm{before})$ is the Work done to bring the object from infinity to a certain position in space, and $W(\mathrm{after})$ the contrary, from position to infinity.

Now, in the case of an electrostatic system, it seems that the Potential Energy is the Work required to bring the charges together (this is what most websites say, example). That would be $$\mathrm{PE} = W(\mathrm{before})$$ Does this mean that $\mathrm{PE} = -W(\mathrm{after})$? That would be for me minus the ability to do Work.

Am I completely missing the idea of $W(\mathrm{after})$ in a electrostatic system?

Best Answer

Remember that every time we talk about work being done, we must know which force that is doing it as well as how the signs are defined.

where $W(begin)$ is the Work done to bring the object from infinity to a certain position in space, and $W(after)$ the contrary, from position to infinity."

What force is doing this work?

That would be some external force pushing them together, which they are resisting (if their signs are equal, which I assume here) giving:

$$PE=W_a=W$$

The electric force between them is repelling and is doing the same amount of negative work! But since it is the system itself that is doing this work, then we put a minus in front:

$$PE=-W_b=-(-W)=W$$

As a rule of thumb, energy into the system is positive. That's why work by external forces is positive, since that could typically bring energy in from the outside. Energy out of the system is negative. That's why work done by the system is given a minus sign, since that could typically mean work spent from the system on something external. But that is very often defined either this or the opposite way in different textbooks.

So it depends

  • first of all on which force is doing the work, and
  • second if all on how the sign on work is defined (is work done on the system positive or negative).

Often this is merely a matter of words, since if the energy content rose then we know without more math that work was done on the system, which in the case of charges means that they are closer together. Then the direction is known and doesn't have to be determined from signs.

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