# The second uniqueness theorem in electrostatics

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• Ahmed1029
In summary, the second uniqueness theorem states that the electric field is uniquely determined if the total charge on each conductor is given and the charge distribution in the regions between the conductors is known.
Ahmed1029
Does the second uniqueness theorem just say that if there is an electric field that satisfies Gauss's law for a surface surrounding each conductor + a surface enclosing all the conductors, it is indeed the true electric field, and no other electric field will satisfy those conditions?

In the link below, the second uniqueness theorem is described as follows.

"The second uniqueness theorem states that the electric field is uniquely determined if the total charge on each conductor is given and the charge distribution in the regions between the conductors is known"

http://teacher.pas.rochester.edu/PHY217/LectureNotes/Chapter3/LectureNotesChapter3.html

I think this conclusion clearly makes sense even without a rigorous mathematical proof.

Last edited:
vanhees71
alan123hk said:
"The second uniqueness theorem states that the electric field is uniquely determined if the total charge on each conductor is given and the charge distribution in the regions between the conductors is known"
I know, but I don't know how to use it when stated like this.
Is what I described above the right way to use it?

I think it's a little different from what you said.

In the figure below, the respective total charges q1, q2, and q3 of conductors s1, s2, and s3 are known, but the charge distribution on each conductor (S1, S2 and S3) is unknown.

Furthermore, the exact charge distribution in all other spaces S0 containing these three conductors is also known.

alan123hk said:
I think it's a little different from what you said.

In the figure below, the respective total charges q1, q2, and q3 of conductors s1, s2, and s3 are known, but the charge distribution on each conductor (S1, S2 and S3) is unknown.

Furthermore, the exact charge distribution in all other spaces S0 containing these three conductors is also known.

that doesn't tell me how to use it or what conditions the electric field must satisfy to be true. All it says is that there is one electric field that is true, it doesn't tell me how to know which one is true. For example I give you 5 different electric fields and ask you which one is the true one, what will you do? What conditions are you going to apply ?

My understanding is that the uniqueness theorem only tells us that the electric field is unique and there cannot be another different electric field. So it can't tell us which electric field is true.

If we want to know what this unique electric field looks like, we must know the exact distribution of charges on conductors s1,s2 and s3.

vanhees71 and Ahmed1029

## What is the second uniqueness theorem in electrostatics?

The second uniqueness theorem in electrostatics states that for a given charge distribution and boundary conditions, there is only one possible electric field that satisfies both the Laplace's equation and the boundary conditions.

## How does the second uniqueness theorem differ from the first uniqueness theorem?

The first uniqueness theorem states that for a given charge distribution, there is only one possible electrostatic potential that satisfies both the Poisson's equation and the boundary conditions. The second uniqueness theorem, on the other hand, deals with the uniqueness of the electric field instead of the potential.

## What are the boundary conditions for the second uniqueness theorem?

The boundary conditions for the second uniqueness theorem are the same as those for the first uniqueness theorem - the potential or electric field must be continuous across the boundary, and the normal component of the electric field must be continuous as well.

## Can the second uniqueness theorem be applied to all charge distributions?

Yes, the second uniqueness theorem can be applied to any charge distribution, as long as the boundary conditions are satisfied. It is a general theorem in electrostatics and does not depend on the specific charge distribution.

## What are some practical applications of the second uniqueness theorem?

The second uniqueness theorem is often used in practical applications involving electrostatics, such as the design of electronic circuits and the analysis of electric fields in conductors. It also helps in solving boundary value problems in electrostatics, which are important in many engineering and scientific fields.

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