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    I want to estimate the energy of point charges over a sphere, however I think I am doing the integral wrong.

    I know the energy is calculated by:  \frac{1}{2}\sum_{i=1}^{N}\sum_{j  =1}^{N} \frac{q_{i}q_{j}}{|p_{i}-p_{j}|}

    Where pi are the position vectors and qi are the charges of the points. Since the charge I am taking is one, and the radius of my sphere is one, when I model the point charges as a uniform charge over the sphere I can get the energy to be:  \frac{1}{2}N^2\frac{1}{16 \pi^2} \int_{S}\int_{S} \frac{1}{|p-p_{0}|}dSdS

    and then taking polar coordinates I get and fixing p_{0} as (0,0,1) I get:

    \frac{N^{2}}{32 \pi^2}\int_0^{2\pi}\int_{0}^{\pi  }\frac{sin(\theta)}{\sqrt{2-2cos(\theta)}}d\theta d\phi

    which cancels to:

    \frac{1}{2}N^2\frac{1}{4\pi}

    However I know it should be \frac{1}{2}N^2, what am I doing wrong??
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    (Original post by adie_raz)
    However I know it should be \frac{1}{2}N^2, what am I doing wrong??
    So you're out by a factor of 4pi, which happens to be the surface area of a sphere of radius 1. So, do you need to take charge density into account? Not that I know anything about this.
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    (Original post by ghostwalker)
    So you're out by a factor of 4pi, which happens to be the surface area of a sphere of radius 1. So, do you need to take charge density into account? Not that I know anything about this.
    I did realise that I was out by the surface area which made me think I was doing it slightly wrong but am definitely close. I shall look into the charge density so thanks you for that.

    I am unsure of my notation as well however, I am not sure if it should in fact be:

     \frac{1}{2}N^2\frac{1}{16 \pi^2} \int_{S} \int_{S} \frac{1}{|p-p_{0}|} d^2{S}d^2{S} ?
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    (Original post by adie_raz)
    I did realise that I was out by the surface area which made me think I was doing it slightly wrong but am definitely close. I shall look into the charge density so thanks you for that.

    I am unsure of my notation as well however, I am not sure if it should in fact be:

     \frac{1}{2}N^2\frac{1}{16 \pi^2} \int_{S} \int_{S} \frac{1}{|p-p_{0}|} d^2{S}d^2{S} ?
    I would have thought:

    \displaystyle\frac{1}{2}N^2\frac  {1}{16 \pi^2} \int_{S} \frac{1}{|p-p_{0}|}\;\mathrm{dS} ?
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    (Original post by ghostwalker)
    I would have thought:

    \displaystyle\frac{1}{2}N^2\frac  {1}{16 \pi^2} \int_{S} \frac{1}{|p-p_{0}|}\;\mathrm{dS} ?
    I think it's because I have fixed the point  p_0 and when I do this I must multiply by the surface area of the sphere. Which sorts my original issue. I shall look into notation and post again when I have an answer. I would agree with you though, it seems I wasn't understanding the notation when I took the notes back in November!
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    Ok so I think it goes as follows:

     \frac{1}{2}N^2\frac{1}{16\pi^2} \int_{S} \int_{S'}\frac{1}{|p-p_{0}|} dS'dS

     = \frac{1}{2}N^2\frac{1}{16\pi^2} \int_{S} \int_{0}^{2\pi}\int_{0}^{\pi} \frac{r^2sin\theta}{\sqrt{2-2cos\theta}} d\theta d\phi dS

     = \frac{1}{2}N^2\frac{4\pi}{16\pi^  2}\int_{S} dS

     = \frac{1}{2}N^2\frac{4\pi}{16\pi^  2} \int_{0}^{2\pi}\int_{0}^{\pi} r^2sin\theta d\theta d\phi

     = \frac{1}{2}N^2\frac{4\pi \times 4\pi}{16\pi^2}

     = \frac{1}{2}N^2
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    (Original post by adie_raz)
    ...
    One thing I would query, is why you have the 16/pi^2 to start with - but I'm not up on the physics behind this.
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    (Original post by ghostwalker)
    One thing I would query, is why you have the 16/pi^2 to start with - but I'm not up on the physics behind this.
    Sorry, I have found that is the charge density I am not up on the physics of it either to be honest, I need to look into it more, but at least I know where I was going wrong now, thanks for the help, always useful to bounce ideas around!
 
 
 
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