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Velocity at any point on a rotating sphere

  1. Dec 5, 2013 #1
    1. The problem statement, all variables and given/known data

    "A uniformly charged solid sphere, of radius R and total charge Q, is centered at the origin and spinning at a constant angular velocity ω about the z axis. Find the current density [itex]\vec{J}[/itex] at any point (r,θ,[itex]\phi[/itex]) within the sphere."
    Problem 5.6(b), p.223, from "Introduction to Electrodynamics" by Griffiths, 4th Edition.

    2. Relevant equations

    [itex]\vec{J}[/itex]=ρ[itex]\vec{v}[/itex].

    3. The attempt at a solution
    The solution is: [itex]\vec{J}[/itex]=ρrωsin(θ)[itex]\hat{\phi}[/itex], with [itex]\vec{v}[/itex]=rωsin(θ)[itex]\hat{\phi}[/itex], and ρ=[itex]\frac{Q}{(4/3)πR^{3}}[/itex].

    Now, my real question: where did the expression for the velocity came from?
    It makes sense that the velocity of any particle on a rotating sphere is given by [itex]\vec{v}[/itex]=rωsin(θ)[itex]\hat{\phi}[/itex], since v is proportional to r, points further away from the origin will be moving faster than those closer to it, ω is the radial velocity, and rω comes from the velocity of a particle in a rotating disk; rωsin(θ), θ being the angle between r and the z-axis, tells us that particles farther away from the z-axis will be moving faster than those closer to it: (assuming we're on the surface) since sin(0)=0 [itex]\Rightarrow[/itex] v=0 is the velocity of the point located at (0,0,R), which is not moving; and sin(π/2)=1 [itex]\Rightarrow[/itex] v is maximum, therefore all the points of any great circle are the fastest.
    And [itex]\hat{\phi}[/itex] just tells us the direction in which the points of the sphere are moving.
    Assuming my interpretation of the velocity equation is correct (correct me if wrong), I ask, where did the equation came from in the first place? I'm sure there has to be an analytical way of deriving such equation, and that it wasn't just a intuitive construction.
    Thank you for your time, and lemme tell you that I'm happy to belong to this community, this being my very first post (I apologize for any issues with the format of my comment); get used to see my username around here. :D

    -Maxarthur

    P.S: Every time that I hit the "Preview post" button, the "2. Relevant equations, ..." subsections get written on my message box. Why could this be? It's annoying having to erase them every time..
     
    Last edited: Dec 5, 2013
  2. jcsd
  3. Dec 5, 2013 #2

    ShayanJ

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    Gold Member

    You can build a sphere by stacking an infinite number of co-centric circles with different radii on each other.Now if the sphere rotates around an axis passing through its poles,all those circles are doing so too with the same angular velocity but because the radii are different,the linear velocity of the points residing on the circles becomes different.[itex] r\sin{\theta} [/itex] is the formula for the radii of those circles which can be easily derived by drawing the shape of what I explained and checking the relation between the radius of the sphere and the radii of those circles.
     
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