sophiecentaur said:
I don't think the photons actually help in this sort of discussion.
It did for me.
The real live 'granularity' of many quantities is not bound to the calculus methods that describe macroscopic relationships. In Calculus, the limit we take is δx→0 and not δx→hf, which is far more specific and arbitrary.
I have no idea what this means.
Anyways,
@another_dude 's question reminded me of an old unsolved problem of mine.
Homework Statement
Given that black body radiation yields values at all wavelengths, guesstimate how many 700 nm photons per time period are emitted from a human body.
constants:
surface area: 1 m^2 (spherical, of course)
temp: 310 K
k_B 1.3806485E-23 J/K Boltzmann constant
h 6.62607E-34 J⋅s Planck constant
c 299792458 m/s speed of light
e 2.71828182845905
λ 700 nm (the limit of human vision on the red end)
Homework Equations
[/B]
Planck's law broken down into bits, as I was sure I would transcribe something wrong
a = 2*h*c^2/λ^5
b = h*c/(λ*k_B * T)
B_λ(λ,T) = a * 1/(e^b - 1) [power output per (area steradian) @ frequency]
Photon energy = hc/λ
The Attempt at a Solution
[/B]
Find B_λ(λ,T) at 700 and 701 nm, ie power output at each point
700 nm --> 1.14e-23 W m-2 sr-1 nm-1
701 nm --> 1.24e-23 W m-2 sr-1 nm-1
Average those two values.
1.19e-23 W m-2 sr-1 nm-1
multiply by the difference in wavelength, 1 nm, yielding bandwidth power output
1.19e-23 W m-2 sr-1
since the subject is 1 m
2, power output will be
1.19e-23 W sr-1
Determine the energy of photons in that range, ie joules/photon @ 700.5 nm
Photon energy = hc/λ =
2.84E-19 joules/photon
I know a watt second = 1 joule ---> watt = joule/sec
so
1.19e-23 W sr-1 =
1.19e-23 joules sec-1 sr-1
therefore, since we want photons/second, we rearrange:
1/2.84e-19 photons/joule * 1.19e-23 joules sec-1 sr-1 = 4.19E-05 photons sec-1 sr-1
which is a very tiny number, so I took the reciprocal, yielding
23866 seconds per photon
which is still too large to recognize, so converted that, and came up with:
A 1 m
2 human @ SOT* will emit a 700 nm photon every 6.6 hours.
So, in answer to the OP's question: although the line is continuous from 700 to 701 nm, the energy of the individual photons in that range limits the output.
Which I am sure is also true if one were to do these types of calculations from 700 nm to 56700 nm.
Which, of course, I did.
Obligatory graph:
Output in this range, per my calculations: 163 watts per steradian (for a 1 m
2 human)
Total output: 525 watts
*SOT = Standard Operating Temperature (98.6°F)