Why Can't the Dirac Delta Function Be a Solution in Electrostatics?

[possible]

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={\delta (r-zk)}
where \delta 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 ?

 

[maajdl]

The Poisson's equation is not satisfied by this potential !

 

[Simon Bridge]

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={\delta (r-zk)}
where \delta 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.

 

[possible]

sorry guys it turned out I was wrong
it doesn't satisfy the poisson's equation

 

[Simon Bridge]

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).

 

[Meir Achuz]

"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.

 

[Simon Bridge]

@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".

 

[Meir Achuz]

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..

 

[Simon Bridge]

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 .
(JIC someone wants to check.)

 

[Meir Achuz]

Other dictionaries include the word 'small' for the distance between, but let's get back to physics.

 

[Simon Bridge]

I agree - the foibles of English use is interesting - but distracting.