Finding temperature from equal radiant power

[BOAS]

Homework Statement

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?

Homework Equations

Q = eσT⁴At

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;

T_s⁴A_s = T⁴_cA_c

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?

 

[phyzguy]

Since it's the same amount of metal, wouldn't you expect the volumes of the sphere and cube to be equal?

 

[BOAS]

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.

 

[Borek]

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

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

Honestly, I have no idea what the problem is.

 

[phyzguy]

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.

 

[BOAS]

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.

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?

 

[Borek]

I don't actually have any numbers for r or d, but the relationship is this;

d = ³√(4/3 πr³)

Looks OK to me.

Am I actually going about this question in a sensible way?

Yes.

Now that you know how d depends on r, you should be able to find how A_c depends on A_s.

 

[BOAS]

So A_c = 6d²

= 6(4/3 πr³)^2/3

T⁴4πr² = T⁴6³√(4/3 πr³)²

T⁴2πr² = T⁴3(4/3 πr³)^2/3

8π³r⁶T¹² = 27(4/3 πr³)²T¹²

8πT¹² = 48T¹²

T = 744 K