How to know what force to plug in for work?

Question

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?

Answer 1

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.

Now, if your question is not related to the potential energy but you actually need to know how much work some device or other external agent will have to do to move a charge against the field then you need to specify HOW do you want this motion to happen. How fast, in how much time, etc. Only then the problem is suficiently defined. The answer will depend on these conditions/restrictions.The force produced by this external agent does not have to be equal in magnitude with the electric force but will be determined by the imposed conditions about the motion.