Photoelectric Effect

2 Planck’s Fundamental Constants

Planck’s constant is independent of his theory of “black body radiation”.

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For large values of T /ν; i.e. for large wavelengths and radiation densities, this equation takes the form

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It is evident that this equation is identical with the one obtained in Sec. 1 from the Maxwellian and electron theories. By equating the coefficients of both formulas one obtains

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or

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An atom of hydrogen weighs 1/N grams = 1.62×10−24 g.

This is exactly the value found by Planck, which agrees with values found by other methods.

I therefore conclude: the greater the energy density and the wavelength of a radiation, the more useful are the theoretical principles.

In small wavelengths and small radiation densities, however, these principles fail us completely.

3. The Entropy of Radiation

Below is Wien’s work.

Let there be radiation taking up volume v.

We assume that the observable properties of the radiation are determined completely when the radiation densities `ρ(ν)`` are given for all frequencies.

Since we can regard radiations of different frequency as separable without doing work or transferring heat the entropy of the radiation can be expressed in the form

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where φ is a function of the variables ρ and ν.

φ can be reduced to a function of only one variable by expressing that the entropy of radiation between reflecting walls is not changed by adiabatic compression.

We won’t go into that however, but investigate right away how the function φ can be obtained from the radiation law of the black body.

In the case of “black body radiation” ρ is such a function of ν that for a given energy the entropy is a maximum, that is, that

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When

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From this it follows that for any choice of δρ as function of ν

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where λ is independent of ν. In the case of blackbody radiation, therefore, ∂ϕ/∂ρ is independent of ν.

The following equation applies when the temperature of a unit volume of blackbody radiation increases by dT

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or, since ∂ϕ/∂ρ is independent of ν.

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is independent of ν:

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Since dE is equal to the heat added and since the process is reversible, the following statement also applies:

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Equating formulas gives:

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This is the law of blackbody radiation. Therefore one can derive the law of blackbody radiation from the function ϕ, and, inversely, one can derive the function ϕ by integration, keeping in mind the fact that ϕ vanishes when ρ = 0.

Limiting (Asymptomic) law for the entropy of monochromatic radiation at low radiation density

Wien’s law is not valid:

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However, for large values of ν/T experiment completely confirms the law.

We shall base our calculations on this formula, keeping in mind that the results will be valid within certain limitations only.

First, we get from this equation:

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and then, using the relation obtained in the preceding section:

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Suppose that we have radiation of energy E, with frequency between ν and ν + dν, enclosed in volume v. The entropy of this radiation is:

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We will limit ourselves to investigating the dependency of the radiation’s entropy on the volume that is occupied.

Let the entropy of the radiation be called S0 when it occupies the volume v0, then we get:

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This equation shows that the entropy of a monochromatic radiation of sufficiently low density varies with the volume in the same manner as the entropy of an ideal gas or a dilute solution.

In the following, this equation will be interpreted in accordance with the principle introduced into physics by Boltzmann, namely that the entropy of a system is a function of the probability its state.