Blackbody radiation derivation of Planck‘s ... A black body is an object that absorbs ... - It accurately describes the spectrum of thermal radiation from ...
Density of States Derivation The density of states gives the number of allowed electron (or hole) states per volume at a given energy. It can be derived from basic quantum mechanics. ... as the effective mass. Rewriting, and noting that the energy of
By coating ordinary paper with layers of gold nanoparticles and other materials, researchers have fabricated flexible paper supercapacitors that exhibit the best performance of any textile-type supercapacitor to date.
Blackbody Radiation and the Loss of Universality: Implications for Planck’s Formulation and Boltzman’s Constant Pierre-Marie Robitaille Department of Radiology, The Ohio State University, 395 W. 12th Ave, Suite 302, Columbus, Ohio 43210, USA E-mail:
Most non-Fickian ergodic transport . theories are based on the effects of long-range temporal correlation due, for example, to solute sorption or preferential pathways.
Using a computer I have found a curios value. And I have a Derivation of the gravitational constant
Derivation of the Wave Equation In these notes we apply Newton’s law to an elastic string, concluding that small amplitude transverse vibrations of the string obey the wave equation.
Spectral Density measurements, this equation is used as the basis for Energy Spectral Density measurements. If we define W(f) to be the Energy Spectral Density of a finite time function…
Energy Density Within Solenoid Energy is stored in the magnetic ﬁeld inside the solenoid. Inductance: L = 0n2A‘ Magnetic ﬁeld: B = 0nI Potential energy: U = 1 2 LI2 = 1 2 0 B2(A‘) Volume of solenoid interior: A‘ Energy density of magnetic ﬁeld: ...
Wien's Displacement Law When the temperature of a blackbody radiator increases, the overall radiated energy increases and the peak of the radiation …
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Planck’s Derivation of the Energy Density of Blackbody Radiation To calculate the number of modes of oscillation of electromagnetic radiation possible in a cavity, consider a one-dimensional box of side L. In equilibrium only standing waves are possible, and these will have nodes at the ends x = 0, L. L = n x n x = 1, 2, 2 and since = c = speed of propagation for all wave motion, = nx c 2L There will be two modes for each trio of integers (one for each dimension) nx, ny, nz because there are two independent polarizations possible. To find the number of modes with frequency between and + d , look at an array of points: z
Each cube has side c . 2L
There is one point per cube of volume c 3, and only positive integers, nx, ny, nz are acceptable. 2L Thus the number of triplets of positive integers is equivalent to the volume of one octant of the space divided by the volume c 3: 2L 2 × 1 × 4π 2 d no. modes of oscillation = 8 = 8π V 2 d between and + d c 3 c3 2L The factor 4π 2 d is the volume of a thin spherical shell; L3 has been replaced by V, the volume of the cavity. It is convenient to express this density of states in terms of other variables: g( ) d = 8 V 2 d g( ) d = 8 V d 4 c3 g( ) d = 8 V 2 d g(p) dp = 8 V p 2 dp 3 hc h3 = h = pc The expressions involving frequency energy , and wavelength are classical physics. If we assume that each mode of oscillation represents a harmonic oscillator, with 1kT each potential 2 and kinetic energy on the average (in accordance with the equipartition theorem), we get the Rayleigh-Jeans law: Energy Energy ≡ u d = 8 kT 2 d or ≡ u d = 8 kT d 4 Volume Volume c3 The divergence of this relation at high frequency or low wavelength was known as the ultraviolet catastrophe. Planck’s new idea was to assume that the possible energies of the oscillators were quantized, i.e., that oscillators of frequency could only have energy n = 0, 1, 2, n = nh where h was a new constant he introduced. Now known as Planck’s constant, it was determined by fitting the theoretical curve to the experimental data. The average energy per oscillator was calculated from the Maxwell-Boltzmann distribution: ∑ n e - n/kT = n ∑ e - n/kT n
Note on black body radiation
The denominator is called the partition function, and is often represented by Z. It is easily evaluated by summing the geometric series: ∞ ∞ Z = ∑ e - n/kT = ∑ e -nx = 1 -x where x = h 1- e kT n=0 n=0 The numerator can then be found from the denominator: ∞ -x ∑ n h e -nx = h - dZ = h e 2 dx -x n=0 1- e and the average energy per oscillator is seen to be h h = ex - 1 e h /kT - 1 Thus the energy per unit volume of the radiation in the cavity is h 3 d 1 u (T) d = 8 or u (T) d = 8 hc d 5 e hc/ kT - 1 c 3 e h /kT - 1 The total energy per unit volume (energy density) is the integral over all frequencies or wavelengths: =
u (T) = 8 h c3
8 kT 3
x 3 dx ex - 1
(hc ) 0 4 The integral is obviously a pure number. It happens to be /15. Thus the energy density in a black body is 8 5 kT 4 u (T) = ≡ aT4 3 15(hc ) This may be thought of as one form of the Stefan-Boltzmann law. [Josef Stefan in 1879 showed experimentally that the flux from a cavity in thermal equilibrium is proportional to the fourth power of the absolute temperature, and Ludwig Boltzmann in 1884 derived this fourth power relation from thermodynamic theory. Until Planck’s work, there was no theoretical method of determining the constants of proportionality.] 0
The flux radiated from the surface of a black body is related to the energy density: 2 1 h 3 = 2 or F = c u = 2 hc 2 h /kT 5 hc/ kT - 1 4 c e -1 e energy where F d = flux = with frequency between and + d . area ⋅ time The total flux, obtained by integrating over all frequencies, is
F = cu 4
F = c u = ac T 4 ≡ T 4 4 4 This is the usual form of the Stefan-Boltzmann law. The constant erg = 5.670 × 10-8 W = 5.670× 10-5 = Stefan-Boltzmann constant. 4 m2 K cm 2 s K4 It is of interest to look at the limits of the Planck distribution. At low frequency or large wavelength, u (T) → 8
u (T) → 8 kT = Rayleigh-Jeans law. 4
Note that Planck’s constant drops out. This is one example of the correspondence principle; as h becomes negligible compared to other quantities in the quantum mechanical law, the result approaches the classical law. At high frequency or small wavelength, 3 u (T) → 8 h e -h /kT and u (T) → 8 hc e -hc/ kT 3 5 c The frequency or wavelength of maximum flux can be found by setting the derivative with respect
Note on black body radiation
equal to zero: 1 0 = dF = 2 h 3 2 h /kT d c e -1
3(h/kT)e h /kT
x which simplifies to 3 = xe with x = h ex - 1 kT The transcendental equation may be solved graphically [graph y = x and y = 3(1 - e-x) and find the nonzero point of intersection] or numerically. The result is x = 2.821 meaning that flux as a function of frequency is a maximum at max = 2.821 kT/h. Maximization of F is similar: ye y 0 = dF ⇒ which yields y = 4.965. = 5 with y = hc d ey - 1 kT Flux as a function of wavelength is a maximum at
2.90 mm ⋅K hc = . 4.965kT T At room temperature T ≅ 290 K, and thermal radiation is a maximum at ≅ 0.01 mm = 10 µm, in the infrared. Fortunately, our eyes are not sensitive to this wavelength. The maximum intensity of the sun’s radiation is at ≅ 500 nm, implying that the sun’s surface temperature is T ≅ 5800 K. The variation of intensity with wavelength of the sun [and other stars] is not exactly that of a black body, but it is rather close. The universal microwave background radiation, peaked at ≅ 1 mm, fits the Planck curve for a black body of T = 2.728 K to great precision. (The deviation, of order 6 parts in 106 is, of course, of great interest.) max