Calculating work on an electron on an equipotential surface

[AdrianMachin]

This is not a homework, but a question formed in my mind after reading my textbook.

Homework Statement

Consider an electron (a charged particle) on a metallic equipotential surface. We know that all the points on the surface are equipotential, thus there will be no force on charged particles on the surface and no tendency to move on their own. We move the electron form point i to point f with our applied force. Find the equation for the work on the electron from the applied force.

Homework Equations

We know from the equation (I) in the attached picture that the work done by the field will be zero because the field lines are prependicular to the surface, thus making the dot product zero.

The Attempt at a Solution

From the equations (II) and (III) we see that the work by the applied force is equal to the changes in the electron's kinetic energy, right? But will our work be zero if ΔK=0? Why? Does it cancel with the work from friction? How?

 

[haruspex]

But will our work be zero if ΔK=0?

Yes.
If ΔK=0 and there is no friction, how great a force was needed/used?

 

[AdrianMachin]

Yes.
If ΔK=0 and there is no friction, how great a force was needed/used?

What do you mean? The magnitude of F_app?
I don't know, that's why I asked this question.

 

[haruspex]

What do you mean? The magnitude of F_app?
I don't know, that's why I asked this question.

Well, do you think 1N would have got it there? 0.1N? 0.00000000001N? Time is immaterial.

 

[AdrianMachin]

Well, do you think 1N would have got it there? 0.1N? 0.00000000001N? Time is immaterial.

I think that makes a paradox because if there are no other forces than the electric force caused by the field which is perpendicular to the surface and the applied force, then there is an acceleration based on Newton's second law, so ΔK cannot be zero...

 

[haruspex]

I think that makes a paradox because if there are no other forces than the electric force caused by the field which is perpendicular to the surface and the applied force, then there is an acceleration based on Newton's second law, so ΔK cannot be zero...

Yes, but it can be arbitrarily small, so effectively zero.

 

[AdrianMachin]

Yes, but it can be arbitrarily small, so effectively zero.

Well, I don't know the answer to the question you asked.

 

[haruspex]

Well, I don't know the answer to the question you asked.

I answered it in post #6. The force is effectively zero.

 

[AdrianMachin]

I answered it in post #6. The force is effectively zero.

Thanks.
Please correct my following conclusions if it is wrong:

The formula for the work done by the applied force on the electron would be $W_{app} = m_ead$, so we must move it with a non-zero acceleration in order to have a non-zero $W_{app}$, and from Newton's second law we'll have $a=\frac {F_{app}-f_k} {m_e}$ so the $F_{app} \neq f_k$ condition must exist.

 

[haruspex]

Thanks.
Please correct my following conclusions if it is wrong:

The formula for the work done by the applied force on the electron would be $W_{app} = m_ead$, so we must move it with a non-zero acceleration in order to have a non-zero $W_{app}$, and from Newton's second law we'll have $a=\frac {F_{app}-f_k} {m_e}$ so the $F_{app} \neq f_k$ condition must exist.

The real world is never ideal. There is always some friction, some extraneous field, whatever. In most academic physics problems you can idealise matters without great consequence, but sometimes it leads to paradoxes.
Yes, if the particle is ever to get to where it is going it must be given a nonzero speed, but there is no time constraint, so there is no nonzero lower limit to the speed. Within any reasonable margin of error, the minimum speed is zero.

 

[AdrianMachin]

Yes, if the particle is ever to get to where it is going it must be given a nonzero speed, but there is no time constraint, so there is no nonzero lower limit to the speed. Within any reasonable margin of error, the minimum speed is zero.

Excuse me, non-zero speed or acceleration?

 

[haruspex]

Excuse me, non-zero speed or acceleration?

Both.