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Gauss's law in action

  1. Apr 15, 2012 #1
    1. The problem statement, all variables and given/known data

    Suppose the electric field in some region is found to be [itex]\vec{E} = kr^{3} \hat{r}[/itex], in spherical coordinates (k is some constant).

    (a) Find the charge density ρ.

    (b) Find the total charge contained in a sphere of radius R, centered at the origin. (Do it two different ways.)

    2. Relevant equations

    3. The attempt at a solution

    (a) Use the formula for the divergence of a vector in spherical basis to get [itex]\nabla . \vec{E} = 5kr^{2}[/itex] so that [itex]\rho = 5k\epsilon_{0}r^{2}[/itex].

    (b) Q is the volume integral of [itex]\rho[/itex] over the volume of the sphere. So, we integrate over d[itex]\phi[/itex], integrate sinθ over dθ, integrate the [itex]\rho[/itex] times [itex]r^{2}[/itex] over dr and multiply the three results. The process gives us 4[itex]\pi \epsilon_{0} R^{5}[/itex].

    Q can also be found using the integral form of Gauss's law, where the surface integral of the electric field is taken with the infinitesimal area, which is R2 sinθ dθ d∅ r, where r is the unit vector in the radial direction. We take the constants out of the integral and integrate 1 over phi and sinθ over theta to obtain the same Q as above.

    Please could you check if the process and the answers are correct?
     
  2. jcsd
  3. Apr 15, 2012 #2
    a) Looks good to me. I get the same.
    b) Is it a hollow spherical shell. Then you can use that the electric field is constant over the surface(fixed r) and only multiply by the area of the sphere(shell) because:
    [itex] \oint \vec{E}\cdot\vec{da} = \oint E\,\text{d}a = E\oint 1\text{d}a = E 4\pi R^2 = Q/\epsilon_0 \Leftrightarrow Q = 4\pi \epsilon_0 kR^5[/itex]
    The dot product [itex]\vec{E}\cdot\vec{da}[/itex] is just da times E cause they point in the same direction.
     
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