A question in Electrostatics

In summary, the question is about the uniqueness theorem and whether a point charge with no electric field around it (except at the point and on the surface of the conductor) satisfies it. The Poisson's equation is not satisfied by this potential.
  • #1
possible
2
0
the question is about the uniqueness theorem (right now I am reading in griffiths book).
the first uniqueness theorem state's that "the solution to laplace's equation in some volume is uniquely determined if the potential is specified on the boundary surface ".
I understand that this gives us the right to use the method of images and to say that there is only one solution there
now here is my question :
if a point charge is placed above an infinite grounded conducting plane then from the first uniqueness theorem there is only one solution to this problem which is the potential due to an electric dipole
but I think there is another solution
which is that the potential is zero everywhere above the conduction surface except at the point where the charge is placed, at that point the potential is infinite
that is
V={[itex]\delta[/itex] (r-zk)}
where [itex]\delta[/itex] is the dirac delta function
and zk is the position of the point charge (0,0,z)
that solution satisfies the boundary conditions and the poisson's equation
so why should that solution be wrong ?
 
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  • #2
The Poisson's equation is not satisfied by this potential !
 
  • #3
possible said:
if a point charge is placed above an infinite grounded conducting plane then from the first uniqueness theorem there is only one solution to this problem which is the potential due to an electric dipole
but I think there is another solution
which is that the potential is zero everywhere above the conduction surface except at the point where the charge is placed, at that point the potential is infinite
that is
V={[itex]\delta[/itex] (r-zk)}
where [itex]\delta[/itex] is the dirac delta function
and zk is the position of the point charge (0,0,z)
that solution satisfies the boundary conditions and the poisson's equation
Really? Please show your working.
 
  • #4
sorry guys it turned out I was wrong
it doesn't satisfy the poisson's equation
 
  • #5
No worries - helps to crunch the numbers.
Although there is a clue in that you seem to have tried for a point charge with no electric field around it (except at the point and on the surface of the conductor).
 
  • #6
"if a point charge is placed above an infinite grounded conducting plane then from the first uniqueness theorem there is only one solution to this problem which is the potential due to an electric dipole"
is also not true with him and. A positive and a negative charge separated by a finite distance is not an electric dipole. It also has higher multiple moments.
 
  • #7
@Meir Achuz:
OK - but have a go writing out a complete description...

An electric dipole can have an electric dipole moment ... can it also have higher order moments?

Intreguing though:
I thought an "electric dipole moment" was a different beastie to an "electric dipole".
 
  • #8
I have always thought of a 'dipole' as having an infinitesimal extent.
I guess some people think it can be of finite size, but then it would also have higher order moments..
 
  • #9
It's understandable - the common English-language use of the term "dipole" refers simply to a physical setup where there are two poles of something. From this use we get the technical term which refers only to the particular term in the multipole expansion of the potential of a distributed group of charges.
This is where I'm usually asked for a reference ... so, for completeness: the common use can be found in the http://www.oxforddictionaries.com/definition/english/dipole [Broken].
(JIC someone wants to check.)
 
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  • #10
Other dictionaries include the word 'small' for the distance between, but let's get back to physics.
 
  • #11
I agree - the foibles of English use is interesting - but distracting.
 

1. What is electrostatics?

Electrostatics is a branch of physics that deals with the study of stationary electric charges and their interactions with each other.

2. What are the fundamental laws of electrostatics?

The fundamental laws of electrostatics are Coulomb's law, which describes the force between two charged particles, and Gauss's law, which relates the electric field to the charge distribution.

3. What is an electric field?

An electric field is a region of space in which an electric charge experiences a force.

4. How is electrostatics used in everyday life?

Electrostatics has many practical applications in our daily lives, such as in the functioning of electronic devices, the creation of static electricity for industrial processes, and the development of medical technologies.

5. What is the role of conductors and insulators in electrostatics?

Conductors are materials that allow electric charges to move freely, while insulators are materials that do not allow the movement of electric charges. These properties play a crucial role in the behavior of electric charges in electrostatics.

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