Discontinuity in an Electric line of force

[vcsharp2003]

Homework Statement

Why is an electric line of force never discontinuous in space except where it meets a charge?

Relevant Equations

$F=k \dfrac {q_1q_2}{d^2}$ where $F$ is the electrostatic force between two point charges $q_1$ and $q_2$ that are a distance $d$ apart (k is the constant of proportionality)

This is a tricky and difficult question for me. I know from reading various textbooks that electric lines of force are always continuous without breaks, but cannot pinpoint a reason for this.

The only reason I can come up is that an electric line of force must always begin and end on charges; this means that if the line became discontinuous then it would not be ending on a charge but in empty space, which is against the concept of an electric line of force.

But, if we consider the case of a single point charge then I know the exact reason. If the line of force became discontinuous then it means electric field is zero in that discontinuous space, which is impossible since according to Coulomb's law there would be an electric field at every point in space. Therefore, an electric line of force cannot be discontinuous in the case of a single point charge.

However, if the scenario was four equal point charges at the four corners of a square, then I have no clue about why line of force cannot be discontinuous. I know that electric field at the center of the square is zero, but the electric lines of force will not suddenly become discontinuous at the center of the square.

 

[haruspex]

That all field lines terminate at a charge depends on the assumption that the net charge of the universe is zero.
A field line is drawn by starting at some arbitrary point and following the field vectors. See https://en.wikipedia.org/wiki/Field_line. If your four charges are at $(\pm 1,\pm 1)$, start at $(-.9, -.9)$. The line will take you to the origin, meeting similar lines from the other three charges. But they don't stop there. Can you figure out where they go?
For an analogy, imagine walking down a hillside towards a saddle point. If you want to keep going downhill, not up the hill in front of you, where do you go?

 

[vcsharp2003]

That all field lines terminate at a charge depends on the assumption that the net charge of the universe is zero.
A field line is drawn by starting at some arbitrary point and following the field vectors. See https://en.wikipedia.org/wiki/Field_line. If your four charges are at $(\pm 1,\pm 1)$, start at $(-.9, -.9)$. The line will take you to the origin, meeting similar lines from the other three charges. But they don't stop there. Can you figure out where they go?
For an analogy, imagine walking down a hillside towards a saddle point. If you want to keep going downhill, not up the hill in front of you, where do you go?

I think the field lines emanating from the four positive point charges would be directed towards the center, but since field lines cannot intersect so the lines coming from each charge will turn away from the center and away from the field lines due to the adjacent two charges.

 

[vcsharp2003]

Can you figure out where they go?

But even if I figure out correctly the path of the field lines as they go towards the center of the square assuming that a line of force does not break in in its path, we still cannot explain why these lines of force cannot have breaks/gaps in it's path.

 

[Steve4Physics]

we still cannot explain why these lines of force cannot have breaks/gaps in it's path.

You probably already know the following, but for completeness:

1. An electric field line (EFL) is a human construct invented to help us conceptualise/represent an electric field.

2. The direction of a tangent to an EFL at a point gives the direction of the electric force experienced by a (small, positive) test charge at that point. That's essentially how we define EFLs.

3. Visually, closely spaced EFLs represent strong fields and widely spaced EFLs represent weak fields. If a region has zero field, we can’t draw any field lines through it.

If a field line started or ended at a point, that would mean the electric field changed from some finite value to zero over an infinitesimally small distance. The electric field would have to be discontinuous at that point – which is unphysical.

 

[vcsharp2003]

You probably already know the following, but for completeness:

1. An electric field line (EFL) is a human construct invented to help us conceptualise/represent an electric field.

2. The direction of a tangent to an EFL at a point gives the direction of the electric force experienced by a (small, positive) test charge at that point. That's essentially how we define EFLs.

3. Visually, closely spaced EFLs represent strong fields and widely spaced EFLs represent weak fields. If a region has zero field, we can’t draw any field lines through it.

If a field line started or ended at a point, that would mean the electric field changed from some finite value to zero over an infinitesimally small distance. The electric field would have to be discontinuous at that point – which is unphysical.

I understand the facts about field lines that you stated. But why would it be unphysical?

I think a line of force must either start from a +ve charge or end at a -ve charge, except when lines come from infinity ( as if they started from infinity) or go to infinity ( as if they are ending at infinity). A break would violate the above mentioned property of line of force since they would be ending neither at a -ve charge or at infinity and the line after the break would be starting neither from a +ve charge or infinity.

But, I am not sure if above explanation sounds valid for why a line of force cannot break in its path.

 

[Steve4Physics]

I understand the facts about field lines that you stated. But why would it be unphysical?

You are asking why a discontinuity (at some arbitrary point in electric field) would be unphysical.

I guess the simple answer is that it is never observed. It would require the electric field gradient at the discontinuity to be infinite! Nature doesn't do that! (Just like you can't accelerate from rest to some speed in zero time.)

I think a line of force must either start from a +ve charge or end at a -ve charge, except when lines come from infinity ( as if they started from infinity) or go to infinity ( as if they are ending at infinity).

That's true for the static case. Note that electric lines of force can be closed loops in dynamic situations, e.g. while a magnetic field is changing.

A break would violate the above mentioned property of line of force since they would be ending neither at a -ve charge or at infinity and the line after the break would be starting neither from a +ve charge or infinity.

But, I am not sure if above explanation sounds valid for why a line of force cannot break in its path.

Your explanation is certainly relevant, but I don't think it addresses the underlying reason - which is that nature has made electric fields continuous (in charge-free regions).

 

[nasu]

The field can be discontinous. And actually is across any surface charge distribution. Of course, a really 2D charge distribution does not exist in reality but the disconinuity is a well accepted fact in EM books and the change is even expressed as a function of the charge density. The potential has to be continous because its gradient is related to the electric field and a discontinuity will mean an infinite field.

Regarding the discontinuity of field line, I don't think this (the discontinuity) is well defined to start with. Does it mean a shap turn of the line or the fact that there is a gap in a specific line? It the later, how de we know that the lines across the gap are the same line? As @Steve4Physics already pointed, the lines are not physical objects.

 

[kuruman]

I understand the facts about field lines that you stated. But why would it be unphysical?

It would be unphysical because it would violate Gauss's law. If you construct a Gaussian surface that encloses the point where the field line disappears without a negative charge to end it, then you have non-zero net flux through the surface while the enclosed charge is zero.

As @Steve4Physics already said and you seem to agree, filed lines are constructs to help us visualize forces on test charges. One can create a set of field lines by moving a test charge charge around in space and drawing small arrows at each point representing the force on the charge at that point. Such a
picture is shown below for two equal positive charges placed on the x-axis. The electric field at the origin is zero and no arrow can be drawn at that point. A zero vector has no direction and no magnitude.

However, you can draw a Gaussian spherical surface (gray circle) and note that that electric flux through the surface is zero which means that you get an equal number of field lines going in as coming out. This is at is should be because there is no enclosed charge. You can shrink the radius of the Gaussian sphere to an arbitrarily small (but non-zero) value and you will still have as many lines going in as coming out. Thus, I think that it is fair to say that the field lines are continuous within the Gaussian surface.

 

[vcsharp2003]

I guess the simple answer is that it is never observed. It would require the electric field gradient at the discontinuity to be infinite!

That makes sense. We never have a quantity suddenly becoming infinite like a vertical rise.

But, if we have a loop for an electric line of force then in going from point A on the loop back to point A such that test charge is always moving quasi- statically on the electric line of force, then in going from A to A there would be some non-zero net work done, which would mean that electric field force is not conservative. We know electric field is always conservative i.e. work done depends only on the initial and final positions.

 

[vcsharp2003]

Does it mean a shap turn of the line or the fact that there is a gap in a specific line?

Sorry, my mistake. I meant a gap.

The potential has to be continous because its gradient is related to the electric field and a discontinuity will mean an infinite field.

That is beautiful. I get that part now and it can be mentioned as a valid explanation. I think you meant the equation below.

$$E = - \dfrac {dV}{dx}$$

 

[vcsharp2003]

It would be unphysical because it would violate Gauss's law. If you construct a Gaussian surface that encloses the point where the field line disappears without a negative charge to end it, then you have non-zero net flux through the surface while the enclosed charge is zero.

Yes, that is very well explained and Gauss's Law cannot be violated; therefore the scenario of a gap in a line of force is not possible.

Thanks for the detailed and excellent explanation. It's very clear now.

 

[Steve4Physics]

We know electric field is always conservative i.e. work done depends only on the initial and final positions.

Not 'always'! If the E-field is induced by a changing B-field, the E-field is non-conservative. E.g. see https://openstax.org/books/university-physics-volume-2/pages/13-4-induced-electric-fields