Calculating Radiation Power of a Ball at 10000K in the Range of 400nm-800nm

[kubajed]

Homework Statement

I need to calculate radiation power of ball in range 400nm-800nm.
T=10000K
d=1um.

Homework Equations

I think I need to use Planck equation and integrate that.
My equation in link: http://imgur.com/XbAUwJ8

The Attempt at a Solution

I look for equations and laws and try to do my equation (above). If this is correct, I ask for help with calculate that.

 

[haruspex]

I think I need to use Planck equation and integrate that

This link http://www.spectralcalc.com/blackbody/integrate_planck.html shows how to integrate it 0 to ∞. Maybe you can adapt it to an arbitrary range.

 

[kubajed]

Thanks. Problem is that I don't know how to calculate integrate.

 

[zexxa]

If you have not learned how to integrate you should consider reading up or watching some explanatory videos on YouTube. I suppose that if the question expects to you integrate for your solution you should have been taught it at some point before.

But if you do know how to integrate, and your only problem is that you are not confident if you did it right, we can help you here if you show us how you did it so we can correct your mistakes (and you can learn from it too!).

 

[kubajed]

I don't know how to integrate. That isn't a homework for all. It's an extra exercise for me. I hope that you will help me with it.
I do calculations in Mathematica. I put that:
Integrate[((2*Pi*6.63*10^-34*(3*10^9)^2)/x^5)*(1/(E^(6.63*10^-34*3*10^9/x*1.38*10^-23*10000))-1)*1.256*10^-11, {x,4*10^-7,8*10^-7}].
I use my equation just multiplied by c/4 and area of ball. Result is -4.31119.
This site: http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/radfrac.html claim that result is 0.00269. Where I am wrong?

 

[haruspex]

I don't know how to integrate. That isn't a homework for all. It's an extra exercise for me. I hope that you will help me with it.
I do calculations in Mathematica. I put that:
Integrate[((2*Pi*6.63*10^-34*(3*10^9)^2)/x^5)*(1/(E^(6.63*10^-34*3*10^9/x*1.38*10^-23*10000))-1)*1.256*10^-11, {x,4*10^-7,8*10^-7}].
I use my equation just multiplied by c/4 and area of ball. Result is -4.31119.
This site: http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/radfrac.html claim that result is 0.00269. Where I am wrong?

This link shows the integral over a finite range http://www.spectralcalc.com/blackbody/inband_radiance.html

 

[kubajed]

Which one of them is that what I am looking for? On this site is calculator too. It claims that result is 6.77561e+07 W/m2/sr. My site: 2.1447e+08 W/m2.

 

[kubajed]

I have other question: how much photons (in visible range) will be registered in detector located 10cm far with diameter 1mm in 100ps?

 

[haruspex]

Which one of them is that what I am looking for? On this site is calculator too. It claims that result is 6.77561e+07 W/m2/sr. My site: 2.1447e+08 W/m2.

Those seem about right for the radiance. It asks for the power. The emitter is a tiny ball. Using https://astrogeology.usgs.gov/tools/thermal-radiance-calculator/, I get more like 2.2E-4W.

(There's something I'm not understanding in this topic. I see radiance quoted as W·sr⁻¹·m⁻². Suppose the ball's surface has a radiance R and radius r. It has a surface area 4πr², so that satisfies the m⁻². What solid angle should I use to satisfy the sr⁻¹? 2π, on the basis that each area element is emitting into a half space? Doesn't really make sense to me.
I note that one of your results quotes sr⁻¹ but the other doesn't, and the ratio is about π.)