# Why electric potential can be evaluated by conservative electric field?

The definition of electric field is following:

The electric field is defined at each point in space as the force per unit charge that would be experienced by a vanishingly small positive test charge if held "stationary" at that point.

The definition of electric potential is:

The electric potential is defined as the amount of work energy needed per unit of electric charge to move this charge from a reference point to the specific point in an electric field.

And it is expressed in by this formula

$$V(r)=\int_{C}E \bullet dr$$ where C is an arbitrary path from reference point to $$r$$.

What i can't understand is from here

ELectric field is force per unit charge experienced by point charge "at rest". But is it reasonable to use electric field for calculating the work needed per unit of electric charge "to move this charge from one point to another point"? The point is how can the force experienced stationary thing be used to calculate for moving one-not stationary.

For example, the coulomb's law explain the force from the stationary charge to stationary charge (not moving charge). $$F=\frac{kQq \vec{r}}{r^2}$$

And electric field due to stationary point charge can be derived from coulomb's law

$$E=\frac{kQ \vec{r}}{r^2}$$

Then the electric potential at r in this electric field is

$$V(r)=\int_{C}E \bullet dr$$ where C is an arbitrary path from reference point to $$r$$.

But this $$E$$ is for stationary one not moving one.

I can't understand why it is reasonable. Or coulomb's law is applicable to explain force from stationary charge to moving charge in uniform velocity??

• "point is how can the force experienced stationary thing be used to calculate for moving one-not stationary" Move it very very very slowly. Do you think there is some limit to how slowly you can move a hypothetical point charge?
– hft
Nov 11, 2023 at 2:35
• Also, the electric field of interest is external to the charge. The charge doesn't act on itself, so you can keep the external field time-independent regardless of how you move the test charge...
– hft
Nov 11, 2023 at 2:37
• @hft, Then the force from stationary charge distribution to moving charge very slowly can be explained by Coulomb's law?
– KHJ
Nov 11, 2023 at 6:15