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Homework Help: Application of Dirac delta

  1. Nov 21, 2011 #1
    1. The problem statement, all variables and given/known data
    We have to give the total charge, dipol and quadrupol moments of a charge constellation, but I seem to be falling at the first hurdle.

    [itex]Q = \frac{1}{4\pi \epsilon_{0}} \int_{vol} \rho(\vec{r}) d^{3}\vec{r}[/itex]

    whereby the charge density of the group of particles is:

    [itex]\rho(\vec{r}) =q\delta(\vec{r} - R\vec{e_{x}}) + q\delta(\vec{r} + R\vec{e_{x}}) + q\delta(\vec{r} - R\vec{e_{y}}) + q\delta(\vec{r} + R\vec{e_{y}}) - 2q\delta(\vec{r} - R\vec{e_{z}}) - 2q\delta(\vec{r} + R\vec{e_{z}})[/itex]

    2. Relevant equations
    I'm using the following property of the delta function:

    [itex] \int_{vol} \delta(\vec{r} - R\vec{e_{x}}) d^{3}\vec{r} = \int_{vol} \delta(x - R) dx \int_{vol} \delta(y)dy \int_{vol} \delta(z)dz = 1 [/itex]

    3. The attempt at a solution

    ok, so I got zero net charge. Which means I don't have a dipol or quadrupol moment either. help!
  2. jcsd
  3. Nov 21, 2011 #2


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    You're right. There's zero net charge. A zero net charge doesn't necessarily mean the dipole and quadrupole moments vanish, however.
  4. Nov 21, 2011 #3
    That's a relief. I still find the delta function a bit confusing.

    ahh, I hadn't realised that about the moments. so I worked out the dipole moment like this:

    [itex]\vec{P} = \int \rho(\vec{r}) \vec{r} \delta(\vec{r} - R\vec{e_{x}}) d^{3} = \Sigma q_{i}\vec{r}_{i}[/itex]

    which in this case was also zero due to cancelling. I'm guessing the quadrupole moment is non-zero, which will be fun :)
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