Why must external force move a charge without accelerating

[Herman Trivilino]

Here we are not told whether this external force is conservative or non conservative.But I think I will suppose it to be non conservative because in our portion /syllabus there are only three types of conservative forces spring force,electric field force and gravitational force.

In the formula

W_nc = ΔPE + ΔKE

you account for the work done in one of two ways. Either as a W_nc or as a ΔPE. The work done by the external force in this question must be accounted for as a W_nc because even if it were a conservative force you have no way of accounting for it as a ΔPE.

If the force is conservative it has a potential energy function, if it isn't it doesn't. Here's another way to look at it.

W_nc = ΔPE + ΔKE
W_nc - ΔPE = ΔKE
W_nc + W_c = ΔKE
W_net = ΔKE

 

[jbriggs444]

The work done by the electrostatic force is path independent. The work done by the other need not be. It could be, for example, that had a different path been taken the total work done might not be zero.

Agreed. That is a reasonable objection. If, for instance, we were to push the charge through the field with our hands, we would not normally describe the force from our hands as "conservative". However, the problem statement does not restrict the set of paths that could be taken. It does require slow movement on any path that actually is taken. So the force applied by our hands must indeed always be equal and opposite to the electrostatic force at every place (and time) where it is applied.

Perhaps I can rephrase as "if the external force is supplied by a force that can be described by a static field then that field must be conservative".

 

[Herman Trivilino]

So the force applied by our hands must indeed always be equal and opposite to the electrostatic force at every place (and time) where it is applied.

Good point. I hadn't thought about that. So in this case the work done by the force applied by the hand actually must be path independent. That, however, doesn't mean that the force is in general path independent. It might be the case, for example, that along some other paths that end in different locations the work done by this particular force might be path dependent. Thus it doesn't qualify as a force for which a potential energy function exists.

Interesting.