# Why is the electric potential continuous

Staff Emeritus
Gold Member
When solving E&M boundary problems, we usually use the condition that the electric potential should be continuous across the boundary. Why is that?

Mentor
Hint: Suppose the electric potential is discontinuous at a certain point. What would the electric force be, on a test charge located at that point?

Discontinuity implies an infinite electric field at the point of the discontinuity. Such fields would thus require an infinite amount of energy to establish and as far as we know thw energy in our universe is finite.

Claude.

Gza
Excellent answers; I was actually pondering the same question a few days back.

harshant
what about point charges? There is discontinuity in potential at origin and although they take an infinite amount of energy they are still used in describing phenomena

pardesi
well one could answer since the gradient of potential has to be the field there otherwise a discontinuity implies so and so field but remember the gradient condtion itself assumes an differenitiable potential hence continious .
the best answer probably is there never has been a case except at the point charge where the potential has been known to blow up or is discontinious.
but if u meant surface charge then it can be very easily proven just take a short line segment across the boundary of the surface charge and calculate the line integral then u can know that as the line get's smaller so does the integrand and hence the potential difference.
note that the integral does exist in the above case though we 'pass' through a 'charge'.remeber we can always integrate function which are discontinious or blow up at finite points the best example would be the greatest integer or mod valued function...
but all these reasons are secondary the most important being experimental verification and nothing else

harshant
although i am wondering whether there is a mathematical proof to show that the potential is continuous everywhere except for point charges.

pardesi
i am wondering whether there is a mathematical proof to show that the potential is continuous everywhere except for point charges.

it's the same as that for surface charges as in my last post take a small line and calculate the line integral

harshant
just one last question, will the potential also be discontinuous on a line charge?

Staff Emeritus
I'm a bit puzzled by not the original question, but by the responses.

First of all, the original question is:

ahrkron said:
When solving E&M boundary problems, we usually use the condition that the electric potential should be continuous across the boundary. Why is that?

I'm surprised that no one actually pointed out clearly that this is actually not true all the time. While the parallel component of the E-field to the surface is continuous across the boundary, the normal component of the E-field need not be continuous all the time. The existence of any surface charge on that boundary will cause a discontinuity in this normal component of the E-field.

Check out here for the detailed derivation using nothing more than Gauss's law.

So no, E-field need not be continuous in all cases.

Zz.

harshant
But the question was about the potential, not the electric field; in fact it was because of this discontinuity of the field that i got confused about the continuity of potential

Staff Emeritus
Oy! You may smack me now.

That's why no one found it puzzling! I need my eyes checked!

Zz.

physicschris
Note on point charges

About the discontinuity at point charges.

Point charges are really just an approximation. Any real charge distribution, even an electron, takes up a finite amount of space. So the potential in any real situation is continuous.

Gold Member
The potential can actually be discontinuous across a dipole layer (in the same way that the electric field is discontinuous across a surface charge). A dipole layer is a surface that has no net charge, but has a dipole moment per area (oriented perpendicular to the surface). Dipole layers can come up in certain situations as a means of approximating two oppositely-charged surfaces, very close together.

However, a dipole layer imposes a very specific condition at the boundary, and so the potential problem is still well-posed.

In continuum mechanics, we can write down a 'jump condition' across a dividing interface, but what I've seen is for the (uncharged) mass, momentum and energy balance equations. I see no reason why one could not write down the same thing when the polarization or potential is taken into account.

Ben- I'd like to know more regarding dipole layers in regards to cell membranes, do you have a good introductory reference?

letitbea
so what's the final conclusion? if i got two dielectrics, the first one in x<0 and the other one in x>0, and a surface charge exists at x=0, will the potential be continuous at x=0 or not?

Crosson
so what's the final conclusion? if i got two dielectrics, the first one in x<0 and the other one in x>0, and a surface charge exists at x=0, will the potential be continuous at x=0 or not?

The potential will be continuous at x = 0, but it will not be differentiable and so the field will have a discontinuity.