Calculating Electric Dipole Moment of Point Charges Along Z-Axis

[joschua]

Hi

I want to calculate the electric dipole moment of point charges along the z-axis with distances a and with the charge distribution

$$\varrho (\vec{x}) = q \delta (\vec{x}) - 2q \delta (\vec{x} - \vec{a}) + q \delta (\vec{x} - 2 \vec{a})$$

and of course $$\vec{a} = a \vec{e}_{z}$$

I did the following:

$$\vec{p} = \int \vec{x}' \varrho (\vec{x'}) d^{3}x'$$

$$= q \int \vec{x}' \delta (\vec{x}) d^{3}x' - 2q \int \vec{x}' \delta (\vec{x} - \vec{a}) d^{3}x' + q \int \vec{x}' \delta (\vec{x} - 2 \vec{a}) d^{3}x'$$

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

$$\int f(x) \delta (x-a) dx = f(a)$$

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? :)

 

[Meir Achuz]

1. All integration variables are x', including in the delta.
2. The f(x) here is just f(x)=x. You are integrating only along the x axis.

 

[joschua]

thanks, I got it