- #1
joschua
- 3
- 0
Hi
I want to calculate the electric dipole moment of point charges along the z-axis with distances a and with the charge distribution
[tex] \varrho (\vec{x}) = q \delta (\vec{x}) - 2q \delta (\vec{x} - \vec{a}) + q \delta (\vec{x} - 2 \vec{a}) [/tex]
and of course [tex]\vec{a} = a \vec{e}_{z} [/tex]
I did the following:
[tex] \vec{p} = \int \vec{x}' \varrho (\vec{x'}) d^{3}x' [/tex]
[tex]= q \int \vec{x}' \delta (\vec{x}) d^{3}x' - 2q \int \vec{x}' \delta (\vec{x} - \vec{a}) d^{3}x' + q \int \vec{x}' \delta (\vec{x} - 2 \vec{a}) d^{3}x' [/tex]
Now I have some questions:
1.) I guess I should write a prime in the arguments of the delta functions. Is this true? (The definition of my electric dipole moment is with prime, the given distribution without but that makes no sense? I should write a prime to all x vectors or no primes. correct?
2.) How to evaluate the integrals further? I know that the delta function is only one at the points of the charges and everywhere else zero but what to do with the x-vectors?
If this would be a normal integral I would do integration by parts, but this makes no sense here.
In general I know the relation that
[tex]\int f(x) \delta (x-a) dx = f(a)[/tex]
but here I have no Function f because x is a vector and I am in 3-d space.
I am confused. Please help me.
edit:
I wanted to post it in Classical Physics and not here. Wrong forum. Sorry... maybe a nice mentor will move it? :)
I want to calculate the electric dipole moment of point charges along the z-axis with distances a and with the charge distribution
[tex] \varrho (\vec{x}) = q \delta (\vec{x}) - 2q \delta (\vec{x} - \vec{a}) + q \delta (\vec{x} - 2 \vec{a}) [/tex]
and of course [tex]\vec{a} = a \vec{e}_{z} [/tex]
I did the following:
[tex] \vec{p} = \int \vec{x}' \varrho (\vec{x'}) d^{3}x' [/tex]
[tex]= q \int \vec{x}' \delta (\vec{x}) d^{3}x' - 2q \int \vec{x}' \delta (\vec{x} - \vec{a}) d^{3}x' + q \int \vec{x}' \delta (\vec{x} - 2 \vec{a}) d^{3}x' [/tex]
Now I have some questions:
1.) I guess I should write a prime in the arguments of the delta functions. Is this true? (The definition of my electric dipole moment is with prime, the given distribution without but that makes no sense? I should write a prime to all x vectors or no primes. correct?
2.) How to evaluate the integrals further? I know that the delta function is only one at the points of the charges and everywhere else zero but what to do with the x-vectors?
If this would be a normal integral I would do integration by parts, but this makes no sense here.
In general I know the relation that
[tex]\int f(x) \delta (x-a) dx = f(a)[/tex]
but here I have no Function f because x is a vector and I am in 3-d space.
I am confused. Please help me.
edit:
I wanted to post it in Classical Physics and not here. Wrong forum. Sorry... maybe a nice mentor will move it? :)
Last edited: