# Is electric potential always continuous in an electrostatic field?

Is electric potential always continuous in an electrostatic field? I mean, does it suffer from discontinuity at any point?

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Potentials are defined up to an additive constant. This means that there is a certain freedom in the value it takes. What I believe is done is that this constant is defined depending on the system of study, so that it is continous. The reason why we do this is contained in this post:

A clue: What relation is there between the electrostatic potential and field?

Potentials are defined up to an additive constant. This means that there is a certain freedom in the value it takes. What I believe is done is that this constant is defined depending on the system of study, so that it is continous. The reason why we do this is contained in this post:

A clue: What relation is there between the electrostatic potential and field?

The scalar potential is continuous everywhere for physical sources, however its first derivative need not be.
The negative first derivative of the scalar potential is the physical field, the electric field. The Electric field is not always continuous due to surface charges.
However one could argue that surface charges are themselves unphysical.

Is electric potential always continuous in an electrostatic field? I mean, does it suffer from discontinuity at any point?
Roughly speaking, yes, it's always continuous in an electrostatic field. However, the electric potential at the point where the electric field source(charge) occupies is not defined.

To make its concept clear, we need to look at the definition of the electric potential.

v = -∫Edl

Obviously, if E vector, the electric field, exists, then we can know the difference of electric potential between the given two points. Next, we let the electric potential at one point be any number we want. So, we get the electric potential at the another one point.

To make its concept clear, we need to look at the definition of the electric potential.

v = -∫Edl

Obviously, if E vector, the electric field, exists, then we can know the difference of electric potential between the given two points. Next, we let the electric potential at one point be any number we want. So, we get the electric potential at the another one point.
What if E vector is discontinuous at one point?

What if E vector is discontinuous at one point?
If you can find an integral path where links the desired point and the point given its electric potential and where E vector is well-defined, then you can calculate the electric potential of the desired point.

So, your question is like "do we always find a path to avoid the undefined-field source position?"

Take a charged spherical metallic ball as an example, I "believe" there is a "good reason" to say that these charges are not "fully" distribute on the surface of this ball.

If there is a "hole" on the surface, then we can link outside and inside through this hole so that the electric potential on/inside the surface is well-defined.

Finally, if this ball example is correct, then I think most of the common cases don't have the problem you cared.

And if, just if, there's a space where is "totally closed" by charges, then it'll need two different reference standard electric potential to define electric potential of every point in the closed space and its outside.

I'm not sure if I'm correct. Hope these thoughts will answer your questions correctly.