# How to know what force to plug in for work?

Suppose I have a positive charge $$+Q$$ at some point, and I want to see how much work I need to do to bring a negative charge $$-q$$ to a distance $$r$$ from that point. The direct calculation is done via the work integral $$W=\int_\infty^r F\cdot dl.$$ I am a little confused here by what to plug in for $$F$$. I have always been told plug in the negative of the electrostatic force experienced by the negative charge, which would lead me to plug in $$qE$$, where $$E$$ is the electric field.

But I feel like this is all operating under the assumption that I want to bring the negative charge to the positive charge at a constant speed. Theoretically, can't I just sit on my hands and let the electrostatic force take care of things? Why do I need to "repel" the attracting force when calculating work?

• Commented Feb 15, 2022 at 18:04

This shows the ambiguity in what you call "the work I need to do". I suppose that you want to calculate this work in relation to potential energy. If this is the case, then you don't need such a thing but just the work done by the electric force. The electric force has no ambiguity and is the same no matter how you move the charge between the two points. The definition of the electrical potential energy is actually only in terms of the work of the electric force. The formula is $$\Delta U=-W_{electric force}=- \int\vec{F}_{electric}\cdot d\vec{r}$$ From this you can see that mathematically you can get rid of the minus sign by taking an imaginary force with equal magnitude and opposite direction to $$F_{electric}$$. You may think that eliminating the minus is worth it but the confusions this may produce (including your question) shows that it should not be used.
In conclusion, forget about this external force and just use the electric force to calculate the work in relation to potential energy or potential (when you use the integral of the electric field, $$\Delta V=- \int\vec{E}\cdot d\vec{r}$$ ) and not of some imagnary field with opposite sign. All decent physics books that you will consult define the potential energy in terms of the force of the field (electric, gravitational) and not some arbitarry external force.