# Calculating work on an electron on an equipotential surface

• AdrianMachin
In summary: The particle must have a non-zero speed in order to move and a non-zero acceleration in order for work to be done by the applied force.
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?

#### Attachments

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AdrianMachin said:
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?

haruspex said:
Yes.
If ΔK=0 and there is no friction, how great a force was needed/used?
What do you mean? The magnitude of Fapp?
I don't know, that's why I asked this question.

AdrianMachin said:
What do you mean? The magnitude of Fapp?
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.

haruspex said:
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...

AdrianMachin said:
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
haruspex said:
Yes, but it can be arbitrarily small, so effectively zero.
Well, I don't know the answer to the question you asked.

AdrianMachin said:
Well, I don't know the answer to the question you asked.
I answered it in post #6. The force is effectively zero.

AdrianMachin
haruspex said:
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.

AdrianMachin said:
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.

haruspex said:
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?

AdrianMachin said:
Excuse me, non-zero speed or acceleration?
Both.

AdrianMachin

## What is the definition of work in terms of electron movement on an equipotential surface?

The work done on an electron on an equipotential surface is the amount of energy required to move the electron from one point to another on the surface, while keeping its kinetic energy constant.

## How is work calculated on an electron on an equipotential surface?

The work is calculated using the equation W = q(Vb - Va), where W is the work done, q is the charge of the electron, and Vb and Va are the potentials at the final and initial points, respectively.

## What is the relationship between work and potential difference for an electron on an equipotential surface?

The work done on an electron is directly proportional to the potential difference between the two points on the equipotential surface. This means that the greater the potential difference, the more work is required to move the electron from one point to another.

## What happens to the work done on an electron on an equipotential surface if the electron follows a closed path?

If the electron follows a closed path on the equipotential surface, the work done will be zero. This is because the potential at the final point will be the same as the potential at the initial point, resulting in a potential difference of zero.

## Can the work done on an electron on an equipotential surface be negative?

Yes, the work done on an electron can be negative if the electron is moving from a point with a higher potential to a point with a lower potential. This means that the electron is releasing energy as it moves along the equipotential surface.

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