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Gauss' Law vector form problem

  1. Feb 1, 2016 #1
    1. The problem statement, all variables and given/known data
    (a) A spherical insulating shell of radius R = 3.00 m has its centre at the origin and carries a surface charge density σ = 3.00 nC/m2. Use Gauss’s law to find the electric field on the x axis at (i) x = 2.00m and (ii) x = 4.00 m. Give you answers in the vector form.

    (b) A point charge q = 250 nC is added to the y axis at y = 2.00 m. Determine the new values of electric field at positions (i) and (ii). Give you answers in the vector form.

    2. Relevant equations
    ∫|E|dA = Q/ε0
    σ = Q/4πr2

    3. The attempt at a solution
    I think I've done the first one my sing a simple gaussian surface arriving at 0 electric field at x = 2.00 as inside the hollow sphere and E = σR20r2 = 190.6 Nc-1 i

    However for part b i get confused as I'm not sure as to wether the charge not being in the centre affects it due to the distribution changing.
    Currently I'm thinking that when x = 2.0m you create a gaussian surface where r=R and the charge enclosed is only that of q. For x = 4 could you do the same as before except the charge enclosed is now σrπR2 + q. That is what i would do if the additional charge was at the origin but as it is not i don't know how to take this into account.
  2. jcsd
  3. Feb 1, 2016 #2
    The first part seems correct to me.
    For the second part you're can only use Gauss's law to determine the ##E-##field when you know the direction of the ##E## field. This is only possible when you have certain perfect symmetries (spherical, infinite plane, infinite cylinder symmetry) which isn't true unless the charge is at the centre. However luckily the ##E## fields obey the superposition principle. So The total field at a point ##r## is simply ##E(r) = E_1(r) + E_2(r)## if you have fields from two charge distributions.
  4. Feb 1, 2016 #3
    So could i just find the electric field of the point charge using kq/r2 and add it to the electric field found by using Gauss law?
  5. Feb 1, 2016 #4
    Yes but remember the fields also have a direction each so you would have to do vector addition.
  6. Feb 1, 2016 #5
    I often get the vector part wrong - so for when x = 4m would E due to q = kq/r2 where r = √(22+42) but that would be in-between the x/y directions. Would I then resolve this is the x direction by taking Esin(θ) where θ=arctan(4/2). And to this term i finally add the field gained in part 1).
  7. Feb 1, 2016 #6
    Involving the trigonometric functions works but is unnecessarily complicated imo. The direction of ##r## is simply ##\hat r## which just is the normed position vector (which you already know in Cartesian coordinates). Drawing a figure often helps when figuring out the direction as well if you are uncertain.
  8. Feb 1, 2016 #7
    Thanks! I'll try and do it that way as i think thats the way we're supposed to! Thank you so much for all of your help.
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