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Electromagnetic Vector Fields (Static)

  1. Mar 20, 2012 #1
    1. The problem statement, all variables and given/known data

    Use the integral form and symmetry arguments to compute the electric field produced by the following charge densities:

    (i) Point charge q, placed at the origin, in 3 dimensions;
    (ii) Point charge q, placed at the origin, in 2 dimensions;
    (iii) Point charge q, placed at the origin, in 1 dimension;
    (iv) Sphere of charge Q, with center at the origin, in 3 dimensions;
    (v) Sphere of charge Q, with center at the origin, in 6 dimensions;

    2. Relevant equations

    [itex]\vec{E}[/itex] and [itex]\vec{B}[/itex] are the electrical and magnetic vector fields, respectively

    Maxwell's Equations (with all constants set to 1):

    [itex]\vec{\nabla}[/itex] [itex]\times[/itex] [itex]\vec{E}[/itex] = -[itex]\partial[/itex][itex]\vec{B}[/itex]/[itex]\partial[/itex]t
    [itex]\vec{\nabla}[/itex] [itex]\times[/itex] [itex]\vec{B}[/itex] = [itex]\partial[/itex][itex]\vec{E}[/itex]/[itex]\partial[/itex]t + [itex]\vec{j}[/itex]
    [itex]\vec{\nabla}[/itex] [itex]\cdot[/itex] [itex]\vec{E}[/itex] = [itex]\rho[/itex]
    [itex]\vec{\nabla}[/itex] [itex]\cdot[/itex] [itex]\vec{B}[/itex] = 0

    3. The attempt at a solution

    No idea how to even set it up. I'm a Math major taking Vector Calculus, Physics is not my cup of tea lol.
     
    Last edited: Mar 21, 2012
  2. jcsd
  3. Mar 21, 2012 #2
    Gauss's law will figure heavily here. See if you can make headway with that.
     
  4. Mar 21, 2012 #3
    The different dimensions are what is confusing me.
     
  5. Mar 22, 2012 #4
    Think of distances in different dimensions. What is the length of a radial vector in 1,2, and 3 dimensions?
     
  6. Mar 22, 2012 #5
    So that would be the vector that connects the origin to whatever point charge I'm using (in whatever dimensions the problem defines)?
     
  7. Mar 23, 2012 #6
    Yes or the distance from the point charge to the field point in question.
     
  8. Mar 25, 2012 #7
  9. Mar 25, 2012 #8
    Here's what I've got so far for part 4:

    Vn = [itex]\frac{R}{n}[/itex] [itex]\times[/itex] 4R2, where n is the dimensions and R is the radius

    ρ(r) = Q/(4πR3/3), where Q is the total charge of the sphere

    [itex]\Rightarrow[/itex] Q = [itex]\int[/itex][itex]\int[/itex][itex]\int[/itex] ρ dV = ρ [itex]\times[/itex] volume = (4πR3/3)ρ

    So then I think I would use the integral form:

    ∫∫ [itex]\vec{E}[/itex](r) [itex]\cdot[/itex] [itex]\hat{n}[/itex] dS

    But [itex]\vec{E}[/itex](r) [itex]\cdot[/itex] [itex]\hat{n}[/itex] = 1, right?

    Then apply the divergence theorem to get (in whatever n dimensions):

    [itex]\underbrace{∫...∫}_{n}[/itex] ([itex]\vec{\nabla}[/itex] [itex]\cdot[/itex] [itex]\vec{E}[/itex]) dV = [itex]\underbrace{∫...∫}_{n}[/itex] ρ dV

    I feel like I'm on the right track. And the approach would be the same for the point charges, except I would be using a radius r > R since the points charge is essentially the limit of the sphere charge as it approaches 0.
     
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