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Electrical Field around Spherical Ball at origin

  1. Nov 12, 2013 #1
    1. The problem statement, all variables and given/known data
    Spherical Ball centered at origin uniform ##\rho## with a radius a. Find E along x-axis.


    2. Relevant equations
    ##E = \frac{\rho}{4\pi\epsilon_0}\int\int\int\frac{r^2*sin\theta}{r_\rho^2} d\phi d\theta dr##


    3. The attempt at a solution
    Evaluate E spherically along the x-axis:
    ##(x_1, 0, 0): r_\rho^2=(x - x_1)^2 + y^2 + z^2=r^2 + x_1^2 - 2*x_1*cos\theta*cos\phi##

    ##r_\rho =\sqrt{r^2 + x_1^2 - 2*x_1*cos\theta*cos\phi}##

    It seems I am missing a component.
     
    Last edited: Nov 12, 2013
  2. jcsd
  3. Nov 12, 2013 #2
    Surely you are! ##r_ρ=\sqrt{r^2+x^2_1−2∗r*x_1∗cosθ∗cosϕ}##

    But you can solve your problem without any troubles using the Gauss theorem:
    $$
    E × 4πx^2 = \frac{ρ × 4/3πa^3}{ε_0}
    $$
     
  4. Nov 12, 2013 #3
    I need to understand this the long way.

    The equation for E I wrote incorrectly
    ##E = \frac{\rho}{4*\pi*\epsilon_0}*\int\int\int\frac{\vec{e_{r_{\rho}}}}{r_{\rho}^2}*r^2*sin^2\theta d\phi d\theta dr##

    How do I find:

    ## \vec{e_{r_\rho}}##
     
    Last edited: Nov 12, 2013
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