Is there a converse of uniqueness theorem

In summary, there is a converse to the uniqueness theorem for circuits in the case of circuit analysis. However, it is a trivial statement and may not hold in all cases. There are two statements that make up the uniqueness theorem, and the converse of the second statement may hold true. Additionally, Gauss's Law in both differential and integral form may also support the converse.
  • #1
pardesi
339
0
is there a converse of uniqueness theorem for circuits have for charged conductors.

or atleast is there such a thing in case of circuit analysis ..
 
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  • #2
Could you state the theorem a bit more precisely for us?
 
  • #3
if in a given volume u know the charge on each conductor(u may notknow the charge distribution) and if u know the charge distribution in space( [tex]\rho[/tex]) then u know everything about that region potential,field...
 
  • #4
So you want to know if the following is true:
If you know the potential, electric field, then you know the charge distribution.
I would say the converse is not true. Afterall, we use the method of images to calculate the potential based on a ficticious distribution of charge.
 
  • #5
If you know the potential and where there are boundaries, then you can determine the charge distribution uniquely.
 
  • #6
@LHarriger
well i don't want to know the charge distribution but the charges


@Parlyne
can u please explain that with an example
 
  • #7
Parlyne said:
If you know the potential and where there are boundaries, then you can determine the charge distribution uniquely.

This is both true and a good point. However, technically this statement is not the converse of the uniqueness theorem (though the techincality is extremely trivial.)

There are actually two statements:

First Uniqueness Theorem: The potential in a volume V is uniquely determined if the charge density throughout the region and the value of the potential V on all boundaries are specified.
Converse: In a volume V, the charge density throughout the region and the value of the potential V on all boundaries are uniquely determined for a given potential V.

I said that this was not true for the reason stated earlier, namely for a given boundary and charge distribution we employ the method of images by defining a fictitious boundary and charge distribution that gives a potential satisfying both the actual and fictitious cases. This is in direct violation of the converse statement. However, I want to change my mind on this. The reason is that when we use the method of images, we are changing our volume of interest. (Infact we must always put our fictitious charge in the expanded volume, otherwise we would be changing our density and would solve Possoin's Equation for the wrong source charge.)
I now am of the mind that the converse should be true. First, if you know the entire potential, then of course you know it on the boundary (like I said, the technicality was trivial.) Second, using Possoin's Equation you can calculate the unique charge density for that potential.

Now that I think about it though, you were probably referring to the second theroem,

Second Uniqueness Theorem Given a volume V that conains conductors of known charge and also contains a known fixed charge density between the conductors, the electric field is uniquely determined. (The region as a whole can be unbounded or surrounded by a conductor).
Converse Given a volume V in which the electric field is known, the charge on conductors inside the volume as well as any charge density between the conductors is uniquely determined.

I would respond on this as well but I have to leave and I want to make sure my response is watertight (unlike last time)
 
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  • #8
so is the converse to the second unique theorem true.also i think 1 and it's converse holds for 1 though the theorem is stronger than the converse i think
 
  • #9
i don't think the terme charge distribution in the converse theorem can be unique because clearly by the second uniqueness theorem once the charge is fixed no matter how it is distributed the field everywhere is fixed provided u know the charge density everywhere.
what i think is true is if u know the field evrywhere then u know the charge density in free space and total charge on each conductor but not how it is distributed
 
  • #10
LHarriger said:
Second Uniqueness Theorem Given a volume V that conains conductors of known charge and also contains a known fixed charge density between the conductors, the electric field is uniquely determined. (The region as a whole can be unbounded or surrounded by a conductor).
Converse Given a volume V in which the electric field is known, the charge on conductors inside the volume as well as any charge density between the conductors is uniquely determined.

I would respond on this as well but I have to leave and I want to make sure my response is watertight (unlike last time)

Gauss's Law in differential form should gaurentee that the converse is true.
 
  • #11
yes got that
gauss law in differential+gauss law in integral
 

What is the converse of uniqueness theorem?

The converse of uniqueness theorem is a mathematical statement that states if a solution to a problem is unique, then it must satisfy certain conditions. This theorem is commonly used in various fields of science, such as physics, engineering, and mathematics.

What is the significance of the converse of uniqueness theorem?

The converse of uniqueness theorem is essential in determining the uniqueness of solutions to problems in various scientific fields. It allows scientists to verify the validity of their solutions and helps in identifying any potential errors or inaccuracies.

How is the converse of uniqueness theorem used in science?

The converse of uniqueness theorem is commonly used in scientific research and experimentation to ensure that the solutions obtained are accurate and reliable. It is also used in the development of mathematical models and theories to validate their uniqueness.

Are there any limitations to the converse of uniqueness theorem?

Like any other mathematical theorem, the converse of uniqueness theorem has its limitations. It may not be applicable to every problem or situation and may require certain conditions to be met for it to be valid. It is crucial to understand these limitations when applying the theorem to scientific problems.

Can the converse of uniqueness theorem be proven?

The converse of uniqueness theorem is a proven mathematical statement and can be derived from other theorems and principles. Its validity has been tested and verified through various scientific experiments and applications, making it a reliable tool in scientific research and problem-solving.

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