How Does the Second Uniqueness Theorem Determine the Electric Field in a Volume?

In summary, the conversation discusses the uniqueness theorem for electric fields in a given volume V surrounded by conductors or infinity with fixed charge density and charge on each conductor. The question is raised about finding the net field outside an uncharged conductor with a cavity containing a point charge, and whether this can be proven using the uniqueness theorem. The conclusion is that the field outside the conductor will be the same as that outside a solid conductor with total surface charge q, and the charge distribution on the conductor must be q for conservation of charge.
  • #1
pardesi
339
0
it states that in a given volume V surrounded by conductors or for that matter infinity if the charge density [tex]\rho[/tex] and the charge on each conductor is fixed then the electric field is uniquely determined in that volume V

Can someone use this find the field in certain situations.
For Example consider this classical situation where in an uncharged conductor has a cavity of arbitrary shape inside it which has a point charge q inside it .The question is to find the net field outside it .
Ofcourse the answer is shielding by the metallic sphere ?
Can someone prove this using the uniqueness theorem .I have a proof in mind but i am unsure of it?
 
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  • #2
What does "net field" mean?
The field outside the conductor will be the same as that outside a solid conductor with total surface charge q. The field outside such a conductor depends on its shape.
By Gauss's law, the surface integral of E is known.
 
  • #3
pam said:
What does "net field" mean?
The field outside the conductor will be the same as that outside a solid conductor with total surface charge q.

this is what i am asking u to prove.
also that the charge q is uniformly distributed
 
  • #4
The surface charge on the conductor has to be q, by conservation of charge.
The surface charge will not be uniform, unless the outer surface is spherical.
 
  • #5
it is a sphere ...
even that doesn't 'prove' that the charge is uniform
 
  • #6
The title of your question was "uniqueness theorem". Use it.
If a spherically symmetry E outside the conductor satisfies all BC, then it is the unique solution. QED.
 

1. What is the Second Uniqueness Theorem?

The Second Uniqueness Theorem is a principle in mathematics that states that if a function satisfies a certain set of conditions, it is the only solution to a given problem. It is commonly used in the field of partial differential equations.

2. What are the conditions for the Second Uniqueness Theorem to apply?

The function must satisfy the Laplace equation, which states that the sum of the second derivatives of the function with respect to each variable is equal to zero. Additionally, the function must also satisfy certain boundary conditions, which specify the behavior of the function at the boundaries of the problem domain.

3. How is the Second Uniqueness Theorem used in practical applications?

The Second Uniqueness Theorem is used in various fields, such as physics and engineering, to find unique solutions to problems involving partial differential equations. It is also used in the study of fluid dynamics and heat transfer.

4. Can the Second Uniqueness Theorem be applied to any problem involving partial differential equations?

No, the Second Uniqueness Theorem can only be applied to problems that satisfy the conditions mentioned earlier. If a function does not satisfy the Laplace equation or the specified boundary conditions, the theorem cannot be used to find a unique solution.

5. Are there any limitations to the Second Uniqueness Theorem?

Yes, the Second Uniqueness Theorem is limited to problems involving linear partial differential equations. It cannot be applied to nonlinear problems, as they may have multiple solutions. Additionally, the theorem only applies to problems with a single solution, and cannot be used to find a range of possible solutions.

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