Planck's Law: Low, and High Frequency Limit

[Destroxia]

Homework Statement

a) Derive the Rayleigh-Jeans distribution by taking the low-frequency limit of Planck's distribution.

b) Derive the Wien distribution by taking the high-frequency limit of Planck's Distribution.

Homework Equations

$u(f) = \frac {8 \pi f^2} {c^3} \frac {hf} {e^{\frac {hf} {k T}}-1}$

The Attempt at a Solution

I think my issue with this problem rather comes from a lack of understanding of what low-frequency, and high-frequency limit means. I read it as, see what happens to the function as the frequency goes very high/low.

So when I do the low-frequency limit, I would imagine, the kT in the exponential would be much larger than hf, therefore making the $e^{\frac {hf} {kT}}$ go to 1, and the whole equation of the form $\frac 0 0$ which is not okay.

When I take a high-frequency limit, I imagine the opposite happens, and we get another indeterminant form of $\frac {\infty} {\infty}$.

The book seems to be doing something like taylor expanding $e^x = 1 + x + ...$ then the negative one in the denominator cancels out, but they still never take a LIMIT as the name low frequency limit implies I should be doing.

Yet for some reason, at high frequency limit, all they do is ignore the 1 in the denominator. I still don't understand why they aren't taking a limit.

I'm just very confused on what the idea is of taking these different limits.

 

[Charles Link]

Homework Statement

a) Derive the Rayleigh-Jeans distribution by taking the low-frequency limit of Planck's distribution.

b) Derive the Wien distribution by taking the high-frequency limit of Planck's Distribution.

Homework Equations

$u(f) = \frac {8 \pi f^2} {c^3} \frac {hf} {e^{\frac {hf} {k T}}-1}$

The Attempt at a Solution

I think my issue with this problem rather comes from a lack of understanding of what low-frequency, and high-frequency limit means. I read it as, see what happens to the function as the frequency goes very high/low.

So when I do the low-frequency limit, I would imagine, the kT in the exponential would be much larger than hf, therefore making the $e^{\frac {hf} {kT}}$ go to 1, and the whole equation of the form $\frac 0 0$ which is not okay.

When I take a high-frequency limit, I imagine the opposite happens, and we get another indeterminant form of $\frac {\infty} {\infty}$.

The book seems to be doing something like taylor expanding $e^x = 1 + x + ...$ then the negative one in the denominator cancels out, but they still never take a LIMIT as the name low frequency limit implies I should be doing.

Yet for some reason, at high frequency limit, all they do is ignore the 1 in the denominator. I still don't understand why they aren't taking a limit.

I'm just very confused on what the idea is of taking these different limits.

The word "limit" is loosely used in this context. What they want you to do is show the behavior of the Planck function at low frequencies and also at high frequencies. They don't actually want a "limit" as f goes to zero or f goes to infinity. Incidentally, the Rayleigh-Jeans distribution works well (fitting experimental data) at low frequencies (e.g. in the mid and far infrared) and the Wien distribution at high frequencies (e.g. in the visible and ultra-violet), but it wasn't until Max Planck came along that a proposed theoretical blackbody energy distribution function agreed throughout the entire spectrum with the experimental results. Note also that the Planck function was the first blackbody curve to also give the result $M=\sigma T^4$ upon integrating it across the entire spectrum. The Planck function is believed to be the correct theoretical expression for the blackbody spectrum, giving results with extremely precise agreement to experimental measurements.

 

[Destroxia]

The word "limit" is loosely used in this context. What they want you to do is show the behavior of the Planck function at low frequencies and also at high frequencies. They don't actually want a "limit" as f goes to zero or f goes to infinity. Incidentally, the Rayleigh-Jeans distribution works well (fitting experimental data) at low frequencies (e.g. in the mid and far infrared) and the Wien distribution at high frequencies (e.g. in the visible and ultra-violet), but it wasn't until Max Planck came along that a proposed theoretical blackbody energy distribution function agreed throughout the entire spectrum with the experimental results. Note also that the Planck function was the first blackbody curve to also give the result $M=\sigma T^4$ upon integrating it across the entire spectrum. The Planck function is believed to be the correct theoretical expression for the blackbody spectrum, giving results with extremely precise agreement to experimental measurements.

Okay I think I understand the low frequency limit, because they only take into account the exponential functions at small values of it's taylor expansion, because the expansion only goes up to the first x term.

But, what is the significance of the high frequency limit? Why can we just ignore the minus 1 next to the exponential in the denominator, and how does this simplify into Wien's Distribution? I don't see it at all.

 

[Charles Link]

Okay I think I understand the low frequency limit, because they only take into account the exponential functions at small values of it's taylor expansion, because the expansion only goes up to the first x term.

But, what is the significance of the high frequency limit? Why can we just ignore the minus 1 next to the exponential in the denominator, and how does this simplify into Wien's Distribution? I don't see it at all.

A google of the Wien distribution will show immediate agreement. They use $\nu$ for frequency instead of f. Typically the parameters in the exponential are 5 or greater for the high f behavior and $e^5-1=e^5$ etc. to a very good approximation. (The minus 1 is insignificant in comparing the U(f) to U(2f) etc. for large f.)