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Is this the correct set up for the electric field?

  1. Oct 9, 2014 #1
    1. The problem statement, all variables and given/known data
    A spherical conductor of radius ##a## carries a charge q and also there is a jelly of constant charge ##rho## per unit volume extending from radius ##a## out to radius ##b##.
    I'm looking to see if I got the correct set up for the electric field of this spherical conductor for all space.

    2. Relevant equations
    ##\oint \vec{E} \cdot d\vec{a}=\frac{Q_{enc}}{\epsilon_0}##

    3. The attempt at a solution
    ##\oint \vec{E} \cdot d\vec{a}=4\pi r^2##
    ##Q_{enc}=\int_a^r 4\pi r'^2 \rho dr'+q=\frac{4\pi}{3}(r^3-a^3)\rho+q##
    So then,
    ##E=\begin{cases}
    0, r<a& \\\
    \rho \frac{(r^3-a^3)}{3r^2 \epsilon_0}+ \frac{q}{4\pi r^2\epsilon_0} \hat{r}, a<r<b\\
    \rho \frac{(b^3-a^3)}{3r^2 \epsilon_0} +\frac{q}{4\pi r^2\epsilon_0} \hat{r}, b\le r
    \end{cases}##
     
  2. jcsd
  3. Oct 9, 2014 #2

    vela

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    Looks good except for the incorrect notation. You should write
    $$\vec{E}=\begin{cases}
    0 & r<a \\
    \left[\rho \frac{(r^3-a^3)}{3r^2 \epsilon_0}+ \frac{q}{4\pi r^2\epsilon_0}\right] \hat{r} & a<r<b \\
    \left[\rho \frac{(b^3-a^3)}{3r^2 \epsilon_0} +\frac{q}{4\pi r^2\epsilon_0}\right] \hat{r} & b\le r
    \end{cases}.$$ The way you wrote it, ##\hat{r}## only multiplies the last term, and you'd be adding a scalar to a vector, which doesn't make sense.
     
  4. Oct 9, 2014 #3
    Yeah I should've added parentheses.Thanks.
     
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