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Electromagnetic fields of a rotating solid sphere: total charge inside

  1. Mar 14, 2012 #1
    1. The problem statement, all variables and given/known data
    A solid sphere of radius a rotates with angular velocity ω[itex]\hat{z}[/itex] relative to an inertial frame K in which the sphere's center is at rest. In a frame K' located at the surface of the sphere, there is no electric field, and the magnetic field is a dipole field with M=M[itex]\hat{z}[/itex] located at the center of the sphere.

    First find the electric and magnetic fields as measured in the K frame and do not assume ωa<<c, then calculate the total charge inside the planet also in the K frame, this time assuming ωa<<c.

    2. Relevant equations

    (i) [itex]\textbf{B}=\frac{3 \hat{r} \left( \hat{r} \bullet \textbf{M} \right) - \textbf{M}} {a^{3}}[/itex]

    (ii) Q[itex]_{enc}[/itex]=[itex]\frac{1}{4π}\int \textbf{E} \bullet \textbf{da}[/itex]

    Also the Lorentz transformation equations to go from E' to E and B' to B (don't want to type...):
    http://en.wikipedia.org/wiki/Lorent...z_transformation_of_the_electromagnetic_field

    3. The attempt at a solution

    I got the transformed electric and magnetic fields, and I want to use (ii) to find the total charge using the electric field I get:

    [itex]\textbf{E}=\frac{Mω} {ca^{2} \sqrt{1-\frac{ω^{2}a^{2}} {c^{2}}sin^{2} \left(θ \right)}} \left(sin^{2}θ \hat{r} - 2sinθcosθ \hat{θ} \right)[/itex]

    BUT I do not know what da would be in this case, since the sphere is rotating in the K frame. Conventionally da is just

    [itex]r dr dθ \hat{r}[/itex]

    EDIT: but that surface element only accounts for part of the electric flux. I guess I'm just not sure. Any insights on this?
     
  2. jcsd
  3. Mar 15, 2012 #2

    tiny-tim

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    hi /flûks/! :smile:
    it doesn't matter that the real sphere is rotating …

    you're integrating over an imaginary sphere! :wink:
    (since you have found E in the stationary frame, you integrate as usual)
     
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