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Gauss law, having problem with spherical load distribution, please help me

  1. Oct 27, 2009 #1
    A region in space containing cargo which is distributed spherically so that the stocking load ρ is given by the equotations:

    ρ = a for r<=R/2
    ρ = 2a(1-r/R) for R/2<=r<=R
    ρ=0 for r>=R

    The total charge Q is 3*10^-17 C, the radius R of the overall distribution is 2*10^-14 m and a is a constant which has dimensions of C / m^3. Determine the constant a as
    function of Q and R and the numerical value. b) Using the
    Gauss's law find an expression for the measure of the electric field as a function of distance r
    from the center of distribution. Do this separately for each of the three areas.
    Make sure that your results agree with the borders of the areas. What fraction of
    total load is contained within the region r<R/2?


    i spent like 3 hours to solve this, but no success
     
  2. jcsd
  3. Oct 27, 2009 #2
    How is charge density, [tex]\rho[/tex], related to total charge [tex]Q[/tex] for a spherical charge? If you can find that relation, the first part is solved by inserting your values of [tex]\rho[/tex] in that relation.

    What is Gauss's Law? More accurately, what is the equation for Gauss's Law? If you know the equation, you should be able to insert the charge density into this and find the answer.
     
  4. Oct 27, 2009 #3
    thanks for the answer

    i know that ρ = Q/(4/3pir^3) right?

    if i take r as R/2 and find ρ, how can i find a? i mean for R/2 we have two different equotations, then

    if i take a gausian surface the Qencl would be Qencl = ρ * V where V = 4/3pir^3 then E would be E = Qencl/eo*Qencl

    i really think it's something like this but im not 100% sure

    ps: this is from the book "university physics by d young"
     
  5. Oct 27, 2009 #4
    You have three areas where the charge density exists:

    [tex]
    \rho=\left[\begin{array}{ll}a &\,\,\,\,\,r<\frac{R}{2} \\ a\left(1-\frac{r}{R}\right)&\,\,\,\,\,\frac{R}{2}\leq r\leq R \\ 0&\,\,\,\,\,r>R\end{array}\right.
    [/tex]

    I would solve the middle one for [tex]a[/tex] in terms of [tex]\rho[/tex]:

    [tex]
    \rho=a\left(1-\frac{r}{R}\right)\rightarrow a=\frac{\rho}{1-\frac{r}{R}}
    [/tex]

    Then replace [tex]\rho[/tex] with your total charge that you have above. The electric field equation you have is wrong, you should have:

    [tex]
    E_r=\frac{\rho}{\varepsilon_0}=\frac{Q_{enc}}{V\varepsilon_0}
    [/tex]
     
  6. Oct 27, 2009 #5
    thank you again, this is actually what i did before, the answer in the book for a question is a = 8Q/5piR^3

    but i keep finding a = 6Q/piR^3
     
    Last edited: Oct 27, 2009
  7. Oct 27, 2009 #6

    gabbagabbahey

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    Why would you do that? :confused:
     
  8. Oct 27, 2009 #7
    hello, gabbagabbahey, what would you do in this case? i'm really stuck, this is like the last exercise i need to solve in order to finish my assignment

    thanks in advance
     
  9. Oct 27, 2009 #8

    gabbagabbahey

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    No, that is only when the charge is uniformly distributed over a sphere of radius 'r'...More generally, the volume charge density is defined according to the equation [itex]dq=\rho dV[/itex], where [itex]dq[/itex] is the amount of charge located in the infinitesimal volume element [itex]dV[/itex]....to get the total charge of any distribution, you need to integrate over all space (or at least a volume that encloses the entire charge distribution):

    [tex]Q=\int_{\text{all space}}\rho(\textbf{r})dV[/tex]
     
  10. Oct 27, 2009 #9
    can you explain me what ρ(r) and "all space" are?

    from the integral i found that Qw = Q*ln(V) where V = 4/3pir^3
     
  11. Oct 27, 2009 #10

    gabbagabbahey

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    Haven't you encountered volume integrals before? Have you taken 2nd year calculus?
     
  12. Oct 27, 2009 #11
    no i dont think so, im studying computer science 1 year, sorry for my bad english i'm from russia
     
  13. Oct 27, 2009 #12

    gabbagabbahey

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    You really need to have taken a multivariable calculus course to understand electrodynamics...what course is this problem for, and what textbook are you studying from? Are you familiar with spherical coordinates?
     
  14. Oct 27, 2009 #13
    it's for physics, this exercise (23.27) is from the book university physics by Hugh D. Young

    where can i find these courses? i really need to understand and solve this by tomorrow

    Thanks
     
  15. Oct 27, 2009 #14

    gabbagabbahey

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    That text has a few examples of volume integrals in spherical coordinates, I suggest you look at them.
     
  16. Oct 27, 2009 #15
    thank you for the answers, i ve studied everything in there

    after some time i found this AWESOME guy http://www.youtube.com/watch?v=nvq_4OFp_IE&NR=1

    explaining different examples, i watched all his videos, and i think i understand what i have to do now

    if i find any problems i ll update this thread

    thanks everyone
     
  17. Oct 27, 2009 #16
    I would've done that because the answer requested the electric field in the three different areas. I was using that as just a suggestion to solve the problem, but maybe I didn't say it was the complete answer but it is for sure a starting point.

    Also, Young's book is an algebra based text (at least the copy I have is), so I'm assuming that he doesn't know anything about integrals, hence not suggestion using one.
     
  18. Oct 27, 2009 #17
    after some search

    because the distribution is not done uniformly

    dq = ρ * dv

    Qencl = integral(ρ*dv)

    Qencl = integral(ρ*4*pi*r^2)dr

    what would ρ be here though? in the above video the man says it's k*r but shouldn't it be Q/(4/3*pi*r^3)?

    im stuck again...

    life can be sucky sometimes :(
     
  19. Oct 27, 2009 #18

    gabbagabbahey

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    You are given [itex]\rho[/itex] in your problem statement...
     
  20. Oct 27, 2009 #19
    yes i used it, but again i cant find the correct answer which is

    a = 8Q/(5*pi*R^3)
     
  21. Oct 27, 2009 #20

    gabbagabbahey

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    How about showing me your steps then...
     
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