# Naked Science Forum

## Non Life Sciences => Physics, Astronomy & Cosmology => Topic started by: lightarrow on 22/08/2006 16:00:33

Title: Wien's displacement law: a (mathematical?) paradox
Post by: lightarrow on 22/08/2006 16:00:33
Wien's displacement law relates a blackbody temperature T with the wavelenght l of its maximum emission: l(max) = a/T, where a is a constant. Let's calculate a.

Spectral radiance as a function of frequency v:

R(v) = [8(pi)h(v/c)^3] * 1/[exp(hv/kT)-1]

as a function of wavelenght l:

R'(l) = (8(pi)hc/l^5) * 1/[exp(hc/lkT)-1].

You can obtain this last formula from the previous one, writing:

Integral(0;+infinity)R(v)dv = Integral(0;+infinity)R'(l)dl

Now we derivate the radiance, to obtain the point of maximum:

dR(v)/dv = 0 --> 1-(1/3)hv/kT = exp(-hv/kT) -->

1-(1/3)hc/lkT = exp(-hc/lkT)  (1)

dR'(l)/dl = 0 -->

1-(1/5)hc/lkT = exp(-hc/lkT) (2)

Solving for l equations (1) and (2) gives two different values of l !

(1) gives (numerically, Mathcad): hc/lkT = 2.82 -->

l(max) = 5.11*10^-3/T

(2) gives (numerically, Mathcad): hc/lkT = 4.97 -->

l(max) = 2.90*10^-3/T

The values taken for computation:
k = 1.38*10^-23 J/°K
h = 6.63*10^-34 J*s
c = 3.00*10^8   m/s

Why two values for a?

Since the blackbody colour depends on l(max), this question could also be put:

which is the real blackbody's colour?
Title: Re: Wien's displacement law: a (mathematical?) paradox
Post by: lightarrow on 23/08/2006 12:03:04
I know, you don't like mathematics.[:(]
Title: Re: Wien's displacement law: a (mathematical?) paradox
Post by: Solvay_1927 on 24/08/2006 01:37:56
I couldn't see any problem with your maths, so I searched Google re: Wien's law and found that you're right - there ARE two different answers.

e.g. see:
http://scienceworld.wolfram.com/physics/WiensDisplacementLaw.html

If you plot the strength of blackbody radiation by frequency, you find that the frequency which produces he maximum emission is v*, say.  And if you plot it by wavelength, you find that the wavelength that produces the maximum emission is l*, say.  But v* times l* is NOT equal to c.

This seems to be an accepted fact.  But I can't find (nor immediately think of) an explanation as to WHY.
Title: Re: Wien's displacement law: a (mathematical?) paradox
Post by: lightarrow on 24/08/2006 07:47:58
Thank you for your replay, Solvay 1927.

I know that physics gives these 2 different results, but isnt'it a paradox?; Why most physics books gives only one of them, that is: l(max) = 2.90*10^-3/T?

How can physicists use only one of them, for example to establish distant stars temperature? See for example:
http://hyperphysics.phy-astr.gsu.edu/hbase/wien.html
quote:
"This relationship is called Wien's displacement law and is useful for the determining the temperatures of hot radiant objects such as stars, and indeed for a determination of the temperature of any radiant object whose temperature is far above that of its surroundings."

Then, the same page states: "It should be noted that the peak of the radiation curve in the Wien relationship is the peak only because the intensity is plotted as a function of wavelength. If frequency or some other variable is used on the horizontal axis, the peak will be at a different wavelength."

So,given T, which is the real blackbody's colour?
Title: Re: Wien's displacement law: a (mathematical?) paradox
Post by: Soul Surfer on 24/08/2006 09:23:33
I haven't worked through the maths but delving back into my memory Wien's equations on radiative enrtgy is an approximation at the high energy end of the spectrum and the Rayliegh equations are an approximation at the low energy end of the radiation spectrum (these go of to infinity as the frequency increases!)and it required the Planck equations (based on the fact that the radiant energy was quantised and the energy of each quantum went up as the frequency increased) This was the start of quantum mechanics to resolve the problem because both the Rayliegy and the Wien equations gave impossible values when you took them outside the area where their approximations were valid.

Learn, create, test and tell
evolution rules in all things
God says so!
Title: Re: Wien's displacement law: a (mathematical?) paradox
Post by: lightarrow on 24/08/2006 10:45:57
quote:
Originally posted by Soul Surfer

I haven't worked through the maths but delving back into my memory Wien's equations on radiative enrtgy is an approximation at the high energy end of the spectrum and the Rayliegh equations are an approximation at the low energy end of the radiation spectrum (these go of to infinity as the frequency increases!)and it required the Planck equations (based on the fact that the radiant energy was quantised and the energy of each quantum went up as the frequency increased) This was the start of quantum mechanics to resolve the problem because both the Rayliegy and the Wien equations gave impossible values when you took them outside the area where their approximations were valid.

As I showed, Wien's displacement law comes from Planck's law, that is from the formula for spectral radiance I wrote:
R(v) = [8(pi)h(v/c)^3] * 1/[exp(hv/kT)-1].
Title: Re: Wien's displacement law: a (mathematical?) paradox
Post by: lightarrow on 23/08/2006 12:03:04
I know, you don't like mathematics.[:(]
Title: Re: Wien's displacement law: a (mathematical?) paradox
Post by: lightarrow on 23/08/2006 12:03:04
I know, you don't like mathematics.[:(]
Title: Re: Wien's displacement law: a (mathematical?) paradox
Post by: Solvay_1927 on 24/08/2006 01:37:56
I couldn't see any problem with your maths, so I searched Google re: Wien's law and found that you're right - there ARE two different answers.

e.g. see:
http://scienceworld.wolfram.com/physics/WiensDisplacementLaw.html

If you plot the strength of blackbody radiation by frequency, you find that the frequency which produces he maximum emission is v*, say.  And if you plot it by wavelength, you find that the wavelength that produces the maximum emission is l*, say.  But v* times l* is NOT equal to c.

This seems to be an accepted fact.  But I can't find (nor immediately think of) an explanation as to WHY.
Title: Re: Wien's displacement law: a (mathematical?) paradox
Post by: lightarrow on 24/08/2006 07:47:58
Thank you for your replay, Solvay 1927.

I know that physics gives these 2 different results, but isnt'it a paradox?; Why most physics books gives only one of them, that is: l(max) = 2.90*10^-3/T?

How can physicists use only one of them, for example to establish distant stars temperature? See for example:
http://hyperphysics.phy-astr.gsu.edu/hbase/wien.html
quote:
"This relationship is called Wien's displacement law and is useful for the determining the temperatures of hot radiant objects such as stars, and indeed for a determination of the temperature of any radiant object whose temperature is far above that of its surroundings."

Then, the same page states: "It should be noted that the peak of the radiation curve in the Wien relationship is the peak only because the intensity is plotted as a function of wavelength. If frequency or some other variable is used on the horizontal axis, the peak will be at a different wavelength."

So,given T, which is the real blackbody's colour?
Title: Re: Wien's displacement law: a (mathematical?) paradox
Post by: Soul Surfer on 24/08/2006 09:23:33
I haven't worked through the maths but delving back into my memory Wien's equations on radiative enrtgy is an approximation at the high energy end of the spectrum and the Rayliegh equations are an approximation at the low energy end of the radiation spectrum (these go of to infinity as the frequency increases!)and it required the Planck equations (based on the fact that the radiant energy was quantised and the energy of each quantum went up as the frequency increased) This was the start of quantum mechanics to resolve the problem because both the Rayliegy and the Wien equations gave impossible values when you took them outside the area where their approximations were valid.

Learn, create, test and tell
evolution rules in all things
God says so!
Title: Re: Wien's displacement law: a (mathematical?) paradox
Post by: lightarrow on 24/08/2006 10:45:57
quote:
Originally posted by Soul Surfer

I haven't worked through the maths but delving back into my memory Wien's equations on radiative enrtgy is an approximation at the high energy end of the spectrum and the Rayliegh equations are an approximation at the low energy end of the radiation spectrum (these go of to infinity as the frequency increases!)and it required the Planck equations (based on the fact that the radiant energy was quantised and the energy of each quantum went up as the frequency increased) This was the start of quantum mechanics to resolve the problem because both the Rayliegy and the Wien equations gave impossible values when you took them outside the area where their approximations were valid.

As I showed, Wien's displacement law comes from Planck's law, that is from the formula for spectral radiance I wrote:
R(v) = [8(pi)h(v/c)^3] * 1/[exp(hv/kT)-1].
Title: Re: Wien's displacement law: a (mathematical?) paradox
Post by: lightarrow on 23/08/2006 12:03:04
I know, you don't like mathematics.[:(]
Title: Re: Wien's displacement law: a (mathematical?) paradox
Post by: Solvay_1927 on 24/08/2006 01:37:56
I couldn't see any problem with your maths, so I searched Google re: Wien's law and found that you're right - there ARE two different answers.

e.g. see:
http://scienceworld.wolfram.com/physics/WiensDisplacementLaw.html

If you plot the strength of blackbody radiation by frequency, you find that the frequency which produces he maximum emission is v*, say.  And if you plot it by wavelength, you find that the wavelength that produces the maximum emission is l*, say.  But v* times l* is NOT equal to c.

This seems to be an accepted fact.  But I can't find (nor immediately think of) an explanation as to WHY.
Title: Re: Wien's displacement law: a (mathematical?) paradox
Post by: lightarrow on 24/08/2006 07:47:58
Thank you for your replay, Solvay 1927.

I know that physics gives these 2 different results, but isnt'it a paradox?; Why most physics books gives only one of them, that is: l(max) = 2.90*10^-3/T?

How can physicists use only one of them, for example to establish distant stars temperature? See for example:
http://hyperphysics.phy-astr.gsu.edu/hbase/wien.html
quote:
"This relationship is called Wien's displacement law and is useful for the determining the temperatures of hot radiant objects such as stars, and indeed for a determination of the temperature of any radiant object whose temperature is far above that of its surroundings."

Then, the same page states: "It should be noted that the peak of the radiation curve in the Wien relationship is the peak only because the intensity is plotted as a function of wavelength. If frequency or some other variable is used on the horizontal axis, the peak will be at a different wavelength."

So,given T, which is the real blackbody's colour?
Title: Re: Wien's displacement law: a (mathematical?) paradox
Post by: Soul Surfer on 24/08/2006 09:23:33
I haven't worked through the maths but delving back into my memory Wien's equations on radiative enrtgy is an approximation at the high energy end of the spectrum and the Rayliegh equations are an approximation at the low energy end of the radiation spectrum (these go of to infinity as the frequency increases!)and it required the Planck equations (based on the fact that the radiant energy was quantised and the energy of each quantum went up as the frequency increased) This was the start of quantum mechanics to resolve the problem because both the Rayliegy and the Wien equations gave impossible values when you took them outside the area where their approximations were valid.

Learn, create, test and tell
evolution rules in all things
God says so!
Title: Re: Wien's displacement law: a (mathematical?) paradox
Post by: lightarrow on 24/08/2006 10:45:57
quote:
Originally posted by Soul Surfer

I haven't worked through the maths but delving back into my memory Wien's equations on radiative enrtgy is an approximation at the high energy end of the spectrum and the Rayliegh equations are an approximation at the low energy end of the radiation spectrum (these go of to infinity as the frequency increases!)and it required the Planck equations (based on the fact that the radiant energy was quantised and the energy of each quantum went up as the frequency increased) This was the start of quantum mechanics to resolve the problem because both the Rayliegy and the Wien equations gave impossible values when you took them outside the area where their approximations were valid.

As I showed, Wien's displacement law comes from Planck's law, that is from the formula for spectral radiance I wrote:
R(v) = [8(pi)h(v/c)^3] * 1/[exp(hv/kT)-1].
Title: Re: Wien's displacement law: a (mathematical?) paradox
Post by: Soul Surfer on 29/08/2006 09:40:08
I noted that but you have to dissect the origins of the equations very carefully to know if there has been an approximation in ththe way they have been formulated.

I am not quite sure why you used that peculier formula containig infinities to change between wavelength and frequency  when the standard equation for any wave motion is wavelength times frequency is velocity.

ie v x l  = c

If you do that the indexin the equation should not change from 3 to 5  this is why your results differ in substitution.

ie v x l  = c

Learn, create, test and tell
evolution rules in all things
God says so!
Title: Re: Wien's displacement law: a (mathematical?) paradox
Post by: lightarrow on 29/08/2006 10:05:07
quote:
Originally posted by Soul Surfer

I noted that but you have to dissect the origins of the equations very carefully to know if there has been an approximation in ththe way they have been formulated.

I am not quite sure why you used that peculier formula containig infinities to change between wavelength and frequency  when the standard equation for any wave motion is wavelength times frequency is velocity.

ie v x l  = c

If you do that the indexin the equation should not change from 3 to 5  this is why your results differ in substitution.

ie v x l  = c
To change from R(v) to R'(l) you can't simply write v = c/l, because dv = -(c/l^2)*dl. You have to equalize the integrals because it's the total energy that is the same.
(You can't write R(v)*dv = R'(l)*dl, at least because if dv is positive then dl is negative!)
Title: Re: Wien's displacement law: a (mathematical?) paradox
Post by: Soul Surfer on 30/08/2006 07:48:36
Sorry I missed the first derivitive index in the second equation but am not sure why yuo used it.

Learn, create, test and tell
evolution rules in all things
God says so!
Title: Re: Wien's displacement law: a (mathematical?) paradox
Post by: Solvay_1927 on 30/08/2006 23:32:23
Ian, that's not a "first derivative index" in the second equation.  The R' in R(l) is just there to distinguish it from the R in R(v) - i.e. to show that the function R'() takes a different format from R().

The reason for using integration (the "peculiar formula containing infinities" as you call it!) is, as Alberto says, because you want to make sure that the TOTAL energy is the same. ( R'(l) has a different SHAPE from R(v), but the total AREA under either curve must be the same. )

Just replacing v/c by l in the first equation does NOT give you "a function in l", i.e. you can replace v/c by l in the right hand side of the first equation, but you can't just change the left hand side from R(v) to R'(l). (I think that would work if "l was linear in v" - i.e. if l = v + constant, say - but not when there's a reciprocal relation between l and v.  Which ties in with Alberto's comment about the derivative being the wrong sign, of course.)
Title: Re: Wien's displacement law: a (mathematical?) paradox
Post by: Solvay_1927 on 30/08/2006 23:34:03
All of which means, of course, that there's still a "paradox" here - WHY are there two different answers for the wavelength?  I'm still very puzzled by that.
Title: Re: Wien's displacement law: a (mathematical?) paradox
Post by: Soul Surfer on 30/08/2006 07:48:36
Sorry I missed the first derivitive index in the second equation but am not sure why yuo used it.

Learn, create, test and tell
evolution rules in all things
God says so!
Title: Re: Wien's displacement law: a (mathematical?) paradox
Post by: Solvay_1927 on 30/08/2006 23:32:23
Ian, that's not a "first derivative index" in the second equation.  The R' in R(l) is just there to distinguish it from the R in R(v) - i.e. to show that the function R'() takes a different format from R().

The reason for using integration (the "peculiar formula containing infinities" as you call it!) is, as Alberto says, because you want to make sure that the TOTAL energy is the same. ( R'(l) has a different SHAPE from R(v), but the total AREA under either curve must be the same. )

Just replacing v/c by l in the first equation does NOT give you "a function in l", i.e. you can replace v/c by l in the right hand side of the first equation, but you can't just change the left hand side from R(v) to R'(l). (I think that would work if "l was linear in v" - i.e. if l = v + constant, say - but not when there's a reciprocal relation between l and v.  Which ties in with Alberto's comment about the derivative being the wrong sign, of course.)
Title: Re: Wien's displacement law: a (mathematical?) paradox
Post by: Solvay_1927 on 30/08/2006 23:34:03
All of which means, of course, that there's still a "paradox" here - WHY are there two different answers for the wavelength?  I'm still very puzzled by that.
Title: Re: Wien's displacement law: a (mathematical?) paradox
Post by: Soul Surfer on 01/09/2006 10:19:13
Sorry I am now getting to understand your nomenclature. Mathematical detail was never my strong point as long as thigs were approximately right enough to work!  :-) I am not familiar with the equations myself as I have not looked at that area since I was at university more than 40 years ago.

My experience with "anomalies" in established equations is that there is usually a mistake in the maths somewhere or a mistake in the units of measurement.

Learn, create, test and tell
evolution rules in all things
God says so!