# Question about equipotential lines and the work done moving along them

• engineeringstudnt
In summary: A force directed towards the center of the particle does work and would cause the particle to move to the center.
engineeringstudnt
Homework Statement
electric potential
Relevant Equations
v=Kq/r
hi guys i have a conceptual question .As you know equipotential surfaces is one on which all point are the same potential there is no work required to move a charge from one point to the other . So my question is how can we change the locotion of a particle without using any force ?

Delta2
The correct formulation is as follows. If a particle moves such that ##V(\boldsymbol r(t))=const## then the force $$\boldsymbol F=-\frac{\partial V}{\partial \boldsymbol r}$$ does no work.

engineeringstudnt
engineeringstudnt said:
Homework Statement:: electric potential
Relevant Equations:: v=Kq/r

hi guys i have a conceptual question .As you know equipotential surfaces is one on which all point are the same potential there is no work required to move a charge from one point to the other . So my question is how can we change the locotion of a particle without using any force ?
You are assuming a particle with mass, yes? And that there is no gravitational field, or the gravitational equipotential coincides with the electrostatic one?
No net work is required. Perhaps the particle is orbiting at constant speed, or you may put some arbitrarily small amount of work into accelerate it then extract that work to stop it.

Steve4Physics and engineeringstudnt
haruspex said:
You are assuming a particle with mass, yes? And that there is no gravitational field, or the gravitational equipotential coincides with the electrostatic one?
No net work is required. Perhaps the particle is orbiting at constant speed, or you may put some arbitrarily small amount of work into accelerate it then extract that work to stop it.
i guess i understand. so we should still use some force even its very very small.if particle is not accelarated initially. right ?

wrobel said:
The correct formulation is as follows. If a particle moves such that ##V(\boldsymbol r(t))=const## then the force $$\boldsymbol F=-\frac{\partial V}{\partial \boldsymbol r}$$ does no work.
thanks sir but can i ask what is "r"?

engineeringstudnt said:
thanks sir but can i ask what is "r"?
I think its the position vector of the particle, in cartesian coordinates ##r=x\hat x+y\hat y+z\hat z## where x the x-coordinate of the particle and ##\hat x## the unit vector of the cartesian system in the x-axis. Also I think the symbolism used it just translates to $$F=-\nabla V=-\frac{\partial V}{\partial x}\hat x-\frac{\partial V}{\partial y}\hat y-\frac{\partial V}{\partial z}\hat z$$.

wrobel and engineeringstudnt
Thank you all :)

As for your main question in equipotential lines the work of the field under consideration is zero, we can do work on the particle with other forces from other fields that don't share the same equipotential curves.

Last edited:
A force normal to the particle's velocity does no work yet would cause a deflection.

Delta2

## 1. What are equipotential lines?

Equipotential lines are imaginary lines that connect points in a field where the potential is the same. They represent areas of equal potential energy.

## 2. How are equipotential lines related to electric fields?

Equipotential lines are always perpendicular to electric field lines. This means that at any point along an equipotential line, the electric field is tangent to the line.

## 3. What does it mean to move along an equipotential line?

Moving along an equipotential line means that the potential energy of the object remains constant. This is because the potential is the same at all points along the line.

## 4. How is work calculated when moving along an equipotential line?

When moving along an equipotential line, no work is done because the potential energy remains constant. This means that the force and displacement are perpendicular to each other, resulting in zero work.

## 5. Can an object move along an equipotential line without any external force?

Yes, an object can move along an equipotential line without any external force because the potential energy remains constant. This means that no work is required to move the object along the line.

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