Plancks law: Relationship between peak wavelength and peak frequency

[ck99]

Homework Statement

What is the relationship between v_max and λ_max and why isn't it just v_maxλ_max = c

Homework Equations

vλ = c

v_max = 5.88 x 10¹⁰T s^-1 K^-1

λ_max = 0.290T^-1 cm K

The Attempt at a Solution

I worked out that B_λ = -dv/dλ B_v

where -dv/dλ = c/λ²

So I thought that λ_max = -dv/dλ v_max but that doesn't seem to work.

Any idea what I have missed? I have been playing around with various rearrangements for a while . . . do I need to do some more calculus?

 

[Dickfore]

How do you determine the peak wavelength and the peak frequency? Are the two conditions satisfied simultaneously?

 

[rude man]

Homework Statement

What is the relationship between v_max and λ_max and why isn't it just v_maxλ_max = c

Homework Equations

vλ = c

v_max = 5.88 x 10¹⁰T s^-1 K^-1

Where did you dig this up?

λ_max = 0.290T^-1 cm K

This is correct. And v_max = c/λ_max unless n > 1 (refraction index).

 

[ck99]

How do you determine the peak wavelength and the peak frequency? Are the two conditions satisfied simultaneously?

Hi and thanks for your reply - I've been working on another problem but I am back on this one now!

Peak wavelength occurs when dB_v / dv = 0 and I already showed that v_max = 5.88 x 10¹⁰T s^-1 K^-1 in part one of the question (taken straight from the lecture notes).

Peak frequency occurs when dB_λ / dλ = 0 and in the question I am given the expression λ_max = 0.290T^-1 cm K as the solution to this.

So aren't these two conditions satisfied when

dB_v / dv = dB_λ / dλ = 0

Is that the same as saying

5.88 x 10¹⁰T s^-1 K^-1 = 0.290T^-1 cm K

I am more confused than ever now as I am pretty sure that makes no sense!

 

[ck99]

Homework Statement

What is the relationship between v_max and λ_max and why isn't it just v_maxλ_max = c

Homework Equations

vλ = c

v_max = 5.88 x 10¹⁰T s^-1 K^-1

Where did you dig this up?

Hi rude man and thanks for your reply. The thing I dug up is from my lecture notes and in the question paper, we work in cgs units in this class, which might be why you don't recognise it?

λ_max = 0.290T^-1 cm K

This is correct. And v_max = c/λ_max unless n > 1 (refraction index).

Hmmm, how can v_max = c/λ_max if my question is "Show that v_maxλ_max ≠ c ? I don't get it . . .

 

[rude man]

Hi rude man and thanks for your reply. The thing I dug up is from my lecture notes and in the question paper, we work in cgs units in this class, which might be why you don't recognise it?



Hmmm, how can v_max = c/λ_max if my question is "Show that v_maxλ_max ≠ c ? I don't get it . . .

That's a good question, and my only answer would be that the index of refraction of the medium is not 1. Example: if you're looking at the radiation from inside a swimming pool, the frequency would be the same as in air but the wavelength would be reduced to c/nf, f = freq in Hz. Water has n = 1.3 or something like that.

 

[rude man]

Hi and thanks for your reply - I've been working on another problem but I am back on this one now!

Peak wavelength occurs when dB_v / dv = 0 and I already showed that v_max = 5.88 x 10¹⁰T s^-1 K^-1 in part one of the question (taken straight from the lecture notes).

Peak frequency occurs when dB_λ / dλ = 0 and in the question I am given the expression λ_max = 0.290T^-1 cm K as the solution to this.

So aren't these two conditions satisfied when

dB_v / dv = dB_λ / dλ = 0

Is that the same as saying

5.88 x 10¹⁰T s^-1 K^-1 = 0.290T^-1 cm K

No, of course not. You don't want to equate a frequency with a wavelength!

What should be true is that λ = c/f or 0.29T^-1 cm-K = c/5.88e10T s^-1K^-1 where c =3e10 cm s^-1. But it isn't ...

I believe some kind of semantic confusion is at the heart of this, as dickfore suggests. If the issue is n > 1 then I calculate n = 1.76.

 

[TSny]

I worked out that B_λ = -dv/dλ B_v

where -dv/dλ = c/λ²

Sub second relation into first to get λ²B_λ = c B_v

So, does B_v have its max when B_λ has its max? Or, does B_v have its max when λ²B_λ has its max?

 

[rude man]

OK, time for me to eat crow again.

Here's the story: the Planck radiation law may be written either as a function of wavelength or as a function of frequency. If you take the wavelength formulation you get λ_max = 0.29T^-1 cm-K whereas if you start with the frequency expression you get hf_max /kT = 2.821439 which corresponds to f_max = 5.88e10T s^-1 K^-1.

As to why these don't agree, the answer is in the formulation of the two versions of the law.
One is not directly derived from the other by λf = c. In fact, as I see it, they contradict each other and I assume both are models of the real thing and therefore inaccurate. Good question for your prof!

If you want to know whichever version of the Planck radiation law you don't have, they can both be found on the Web.

Ya live and learn!

 

[ck99]

OK, time for me to eat crow again.

Here's the story: the Planck radiation law may be written either as a function of wavelength or as a function of frequency. If you take the wavelength formulation you get λ_max = 0.29T^-1 cm-K whereas if you start with the frequency expression you get hf_max /kT = 2.821439 which corresponds to f_max = 5.88e10T s^-1 K^-1.

As to why these don't agree, the answer is in the formulation of the two versions of the law.
One is not directly derived from the other by λf = c. In fact, as I see it, they contradict each other and I assume both are models of the real thing and therefore inaccurate. Good question for your prof!

If you want to know whichever version of the Planck radiation law you don't have, they can both be found on the Web.

Ya live and learn!

I showed in my first post that I worked out how to go from one form to the other. They are not "models" and they are both accurate.

 

[ck99]

Sub second relation into first to get λ²B_λ = c B_v

So, does B_v have its max when B_λ has its max? Or, does B_v have its max when λ²B_λ has its max?

Hi TSny and thanks for your reply.

I think I want to show that

dB_v / dv = 0 when dλ² c^-1 B_λ /dλ = 0

However, using a similar substitution method I used to derive dB_v /dv = 0, I just end up getting exactly the same result for both expressions! So that equation is true, which again makes me think that the conversion factor should just be λ²/c but as soon as I put a value in for T and actually compare numbers that doesn't work. What am I missing?

 

[TSny]

I think I want to show that
dB_v / dv = 0 when dλ² c^-1 B_λ /dλ = 0

We have cB_v = λ²B_λ where v on the left is related to λ on the right by c = vλ. B_v will take on its maximum value when λ²B_λ has its maximum value. So, the value of λ that corresponds to the value of v_max (where Bv takes on its max) would be obtained from B_λ by the condition d(λ²B_λ)/dλ = 0. Show that this leads to dB_λ/dλ = -(2/λ)B_λ.

So, the value of λ that corresponds to v_max will not be the value of λ that makes B_λ a maximum, but to the value of λ that makes dB_λ/dλ = -(2/λ)B_λ. So, dB_λ/dλ will be negative at this point, which means that λ will be on the "downhill" side of the peak of B_λ. In other words, the value of λ that corresponds to v_max will be greater than λ_max (where dB_λ/dλ = 0).