Learn Wien's Displacement Law Derivation with Plainmath's Expert Help

The velocity function is for a particle moving along a line. What is the displacement (net distance covered) of the particle during the time interval [-3,6]?

How do velocity and acceleration differ?

A passenger plane made a trip to Las Vegas and back. On the trip there it flew 432 mph and on the return trip it went 480 mph. How long did the trip there take if the return trip took nine hours?

The position of an object moving along a line is given by . What is the speed of the object at ?

A person observes fireworks displays from the safe distance of 0.750 meters. Assuming that the sound travels at 340 meters per second in air. What is the time between the person seeing the fireworks and hearing the explosion?

What is the speed of a rocket that travels 9000 meters in 12.12 seconds?

The distance between the earth and the moon is about 384,000 km. calculate the time it takes for light to travel from the moon to earth?

The velocity of the particle is given by v=3t^2+2t in m/s. The acceleration and displacement of the particle as a function of time respectively are...

The transverse displacement y(x, t) of a wave on a string is given by $y(x,t)=e^{(a{x}^2+b{t}^2+2\sqrt{ab}xt)}$. This represent:
A) wave moving in +x direction with speed $\sqrt{\frac{a}{b}}$
B) wave moving in +x direction with speed $\sqrt{\frac{b}{a}}$
C) standing wave of frequency $\sqrt{b}$
D) standing wave of frequency $\frac{1}{\sqrt{b}}$

A husband and wife take turns pulling their child in a wagon along a horizontal sidewalk. Each exerts a constant force and pulls the wagon through the same displacement. They do the same amount of work, but the husband's pulling force is directed ${58}^o$ above the horizontal, and the wife's pulling force is directed ${38}^o$ above the horizontal. The husband pulls with a force whose magnitude is 60 N. What is the magnitude of the pulling force exerted by his wife?

Can displacement be negative?

The displacement of a particle at t = 0.350 s is given by the expression x = (3.00 m) \cos(6.00xt), where x is in meters and is in seconds (a) Determine the frequency and period of the motion Hz s (b) Determine the amplitude of the motion m (c) Determine the displacement of the particle at t = 0.350 s. m (d) At what time after t=0 does it first reach equillibrium? (e) At what time after t=0 does it reach zero velocity for the first time? (f)What total distance does it travel during the first period?

Landauer's principle vs Wien's displacement law
Can we argue based on Landauer's principle that if one bit information is changed inside a blackbody, the total radiated energy should be at least or in order of kTln2? If it is so, can we also argue that this energy should be distributed over all the modes of the cavity? Furthermore, can it also be argued that this contradicts with the Wien's displacement law which says the total energy should be in the order of $k\frac{T^4}{h^4}$ (as integrated using Mathematica 8.0)?

Wien's Displacement and Rayleigh-Jeans Law from Planck's Law
I've derived Planck's Law for frequency from his law for wavelength, and I got this:
$u(f)=\frac{8\mathrm{\pi <!--\; \pi \; -->}{f}^2}{c^3}\frac{hf}{e^{\frac{hf}{kT}}-<!--\; -\; -->1}$
I just have a quick question about this. This problem says I need to find the low-frequency limit (which will lead to the Rayleigh-Jeans), and I need to take the high-frequency limit, which is supposed to lead to Wien's distribution.
The thing is I have no idea what the high, or low frequency limit even means, I've looked in my book, I've looked online, I'm not sure exactly what it means.

How do we know that dark matter is dark?
How do we know that dark matter is dark, in the sense that it doesn't give out any light or absorb any? It is impossible for humans to be watching every single wavelength. For example, what about wavelengths that are too big to detect on Earth?

How do you change Planck's law from frequency to wavelength?
I have to derive Wien's displacement law by using Planck's law. I've tried but I come to a unsolvable equation (well I can't solve it) anywhere I look online it comes to the same conclusion, you need to solve an equation involving transcendentals which I have no idea what they are nor any of the math classes required for this physics course teach it. Once that equation is solved, the next steps are quite simple, by using the solution.
Maybe my professor just wants us to google the solution to the equation and used it to keep solving the problem?
anyway, what really bothers me is this:
My book shows Planck's law in terms of frequency
$B_{\mathrm{\nu <!--\; \nu \; -->}}(T)=\frac{2h{\mathrm{\nu <!--\; \nu \; -->}}^3}{c^2}\frac{1}{e^{h\mathrm{\nu <!--\; \nu \; -->}/(k_{B}T)}-<!--\; -\; -->1}$
but to do the problem I need it in terms of wavelength
$B_{\mathrm{\lambda <!--\; \lambda \; -->}}(T)=\frac{2h{c}^2}{{\mathrm{\lambda <!--\; \lambda \; -->}}^5}\frac{1}{e^{hc/(\mathrm{\lambda <!--\; \lambda \; -->}{k}_{B}T)}-<!--\; -\; -->1}$
And $v=c/\mathrm{\lambda <!--\; \lambda \; -->}$, so by that then it can be said that the exponent of $\mathrm{\lambda <!--\; \lambda \; -->}$ would be 3 when it is expressed in terms of lambda (plus $c$ would have an exponent of 1). But Wikipedia says it is 5 and not 3. It multiplies by the derivative of $\mathrm{\nu <!--\; \nu \; -->}$, but I don't get why. I'm just substituting, not differentiating, no need to use the chain rule.

What causes hot things to glow, and at what temperature?
I have an electric stove, and when I turn it on and turn off the lights, I notice the stove glowing.
However, as I turn down the temperature, it eventually goes away completely. Is there a cut-off point for glowing?
What actually is giving off the light? Does the heat itself give off the light, or the metal?

Relationship between Wien's law and Stefan-Boltzmann's law
I'm studying Quantum mechanics by Bransden and Joachain and in the introduction chapter it says:
Wien showed that the spectral distribution function had to be on the form
$\mathrm{\rho <!--\; \rho \; -->}(\mathrm{\lambda <!--\; \lambda \; -->},T)={\mathrm{\lambda <!--\; \lambda \; -->}}^{-<!--\; -\; -->5}f(\mathrm{\lambda <!--\; \lambda \; -->}T)$
where $f(\mathrm{\lambda <!--\; \lambda \; -->}T)$ is a function of the single variable $\mathrm{\lambda <!--\; \lambda \; -->}T$. It is a simple matter to show that Wien's law includes Stefan-Boltzmann law $R(T)=\mathrm{\sigma <!--\; \sigma \; -->}{T}^4$
One of the exercises is to show this and I cannot understand how to.
This is what I've tried:
The relationship between spectral emittance and spectral distribution is
$\mathrm{\rho <!--\; \rho \; -->}(\mathrm{\lambda <!--\; \lambda \; -->},T)=\frac{4}{c}R(\mathrm{\lambda <!--\; \lambda \; -->},T),$
where c is the speed of light, which inserted in the above equation gives
$R(\mathrm{\lambda <!--\; \lambda \; -->},T)=\frac{c}{4}{\mathrm{\lambda <!--\; \lambda \; -->}}^{-<!--\; -\; -->5}f(\mathrm{\lambda <!--\; \lambda \; -->}T).$
Now, the total spectral emittance is the integral of $R$ over all wavelengths so
$R(T)=\frac{c}{4}\underset{0}{\overset{\mathrm{\infty <!--\; \infty \; -->}}{\int <!--\; \int \; -->}}{\mathrm{\lambda <!--\; \lambda \; -->}}^{-<!--\; -\; -->5}f(\mathrm{\lambda <!--\; \lambda \; -->}T)d\mathrm{\lambda <!--\; \lambda \; -->}$
This is where I'm stuck. Can anyone help me figure this out?

Wien's displacement law in frequency domain
When I tried to derive the Wien's displacement law I used Planck's law for blackbody radiation:
$I_{\mathrm{\nu <!--\; \nu \; -->}}=\frac{8\mathrm{\pi <!--\; \pi \; -->}{\mathrm{\nu <!--\; \nu \; -->}}^2}{c^3}\frac{h\mathrm{\nu <!--\; \nu \; -->}}{e^{h\mathrm{\nu <!--\; \nu \; -->}/k_{b}T}-<!--\; -\; -->1}$
Asking for maximum:
$\frac{d{I}_{\mathrm{\nu <!--\; \nu \; -->}}}{d\mathrm{\nu <!--\; \nu \; -->}}=0:0=\frac{\mathrm{\partial <!--\; \partial \; -->}}{\mathrm{\partial <!--\; \partial \; -->}\mathrm{\nu <!--\; \nu \; -->}}(\frac{{\mathrm{\nu <!--\; \nu \; -->}}^3}{e^{h\mathrm{\nu <!--\; \nu \; -->}/k_{b}T}-<!--\; -\; -->1})=\frac{3{\mathrm{\nu <!--\; \nu \; -->}}^2(e^{h\mathrm{\nu <!--\; \nu \; -->}/k_{b}T}-<!--\; -\; -->1)-<!--\; -\; -->{\mathrm{\nu <!--\; \nu \; -->}}^{3}h/k_{b}T\cdot <!--\; \cdot \; -->e^{h\mathrm{\nu <!--\; \nu \; -->}/k_{b}T}}{(e^{h\mathrm{\nu <!--\; \nu \; -->}/k_{b}T}-<!--\; -\; -->1{)}^2}$
It follows that numerator has to be $0$ and looking for $\mathrm{\nu <!--\; \nu \; -->}>0$
$3(e^{h\mathrm{\nu <!--\; \nu \; -->}/k_{b}T}-<!--\; -\; -->1)-<!--\; -\; -->h\mathrm{\nu <!--\; \nu \; -->}/k_{b}T\cdot <!--\; \cdot \; -->e^{h\mathrm{\nu <!--\; \nu \; -->}/k_{b}T}=0$
Solving for $\mathrm{\gamma <!--\; \gamma \; -->}=h\mathrm{\nu <!--\; \nu \; -->}/k_{b}T$:
$3(e^{\mathrm{\gamma <!--\; \gamma \; -->}}-<!--\; -\; -->1)-<!--\; -\; -->\mathrm{\gamma <!--\; \gamma \; -->}{e}^{\mathrm{\gamma <!--\; \gamma \; -->}}=0\to <!--\; \to \; -->\mathrm{\gamma <!--\; \gamma \; -->}=2.824$
Now I look at the wavelength domain:
$\mathrm{\lambda <!--\; \lambda \; -->}=c/\mathrm{\nu <!--\; \nu \; -->}:\mathrm{\lambda <!--\; \lambda \; -->}=\frac{hc}{\mathrm{\gamma <!--\; \gamma \; -->}{k}_b}\frac{1}{T}$
but from Wien's law $\mathrm{\lambda <!--\; \lambda \; -->}T=b$ I expect that $hc/\mathrm{\gamma <!--\; \gamma \; -->}{k}_b$ is equal to $b$ which is not:
$\frac{hc}{\mathrm{\gamma <!--\; \gamma \; -->}{k}_b}=0.005099$, where $b=0.002897$
Why the derivation from frequency domain does not correspond the maximum in wavelength domain?
I tried to justify it with chain rule:
$\frac{dI}{d\mathrm{\lambda <!--\; \lambda \; -->}}=\frac{dI}{d\mathrm{\nu <!--\; \nu \; -->}}\frac{d\mathrm{\nu <!--\; \nu \; -->}}{d\mathrm{\lambda <!--\; \lambda \; -->}}=\frac{c}{{\mathrm{\nu <!--\; \nu \; -->}}^2}\frac{dI}{d\mathrm{\nu <!--\; \nu \; -->}}$
where I see that $c/{\mathrm{\nu <!--\; \nu \; -->}}^2$ does not influence where $d{I}_{\mathrm{\lambda <!--\; \lambda \; -->}}/d\mathrm{\lambda <!--\; \lambda \; -->}$ is zero.

Uses of Wien's law of displacement
Wiens's displacement law says
${\lambda }_{\text{max}}T=\text{a constant}$
So if I have the ${\lambda }_{\text{max}}$, I can find the temperature of a star. But if I have the temperature, is there any point in calculating ${\lambda }_{\text{max}}$? What information does that give us of the star, besides temperature?