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Finding temperature from equal radiant power

  1. Nov 11, 2013 #1
    1. The problem statement, all variables and given/known data

    A solid sphere has a temperature of 785 K. The sphere is melted down and recast into
    a cube that has the same emissivity and emits the same radiant power as the sphere.
    What is the cube's temperature?

    2. Relevant equations

    Q = eσT4At

    3. The attempt at a solution

    I know that the radiant power and emmissivity of the two objects are the same and σ is a constant so I can say that;

    Ts4As = T4cAc

    Subscript s and c for sphere and cube.

    I know the sphere's temperature and can express it's area as 4∏r2 and the area of the cube can be expressed as 6r2. Where r represents one side.

    And now, I don't know what to do. Is there a relationship that links the volume of a sphere to the size of cube that can be made from it?
     
  2. jcsd
  3. Nov 11, 2013 #2

    phyzguy

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    Science Advisor

    Since it's the same amount of metal, wouldn't you expect the volumes of the sphere and cube to be equal?
     
  4. Nov 11, 2013 #3
    Yes, that was a mistake.

    What I meant to say was is there a relationship between the sphere and the cube that can be used to find surface area.

    For a sphere of radius r, i'm pretty sure that only one cube can be constructed.
     
  5. Nov 11, 2013 #4

    Borek

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    Staff: Mentor

    Yes, in this case there exist a relationship - their volumes are equal. Use that fact to find how rc depends on rs.

    Honestly, I have no idea what the problem is.
     
  6. Nov 11, 2013 #5

    phyzguy

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    Science Advisor

    So if you know the radius of the sphere (call it r), you can calculate the side of the cube (call it d so you don't get confused) that has the same volume, right? So then you can calculate As and Ac from the formulae you gave before.
     
  7. Nov 11, 2013 #6
    I don't actually have any numbers for r or d, but the relationship is this;

    d = 3√(4/3 πr3)

    T44πr2 = T463√(4/3 πr3)2

    Am I actually going about this question in a sensible way?
     
    Last edited: Nov 11, 2013
  8. Nov 11, 2013 #7

    Borek

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    Staff: Mentor

    Looks OK to me.

    Yes.

    Now that you know how d depends on r, you should be able to find how Ac depends on As.
     
  9. Nov 11, 2013 #8
    So Ac = 6d2

    = 6(4/3 πr3)2/3

    T44πr2 = T463√(4/3 πr3)2

    T42πr2 = T43(4/3 πr3)2/3

    3r6T12 = 27(4/3 πr3)2T12

    8πT12 = 48T12

    T = 744 K
     
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