Planck's law

In physics, Planck's law (also Planck radiation law) describes the spectral density of electromagnetic radiation emitted by a black body in thermal equilibrium at a given temperature T, when there is no net flow of matter or energy between the body and its environment.

At the end of the 19th century, physicists were unable to explain why the observed spectrum of black-body radiation, which by then had been accurately measured, diverged significantly at higher frequencies from that predicted by existing theories. In 1900, German physicist Max Planck heuristically derived a formula for the observed spectrum by assuming that a hypothetical electrically charged oscillator in a cavity that contained black-body radiation could only change its energy in a minimal increment, E, that was proportional to the frequency of its associated electromagnetic wave. While Planck originally regarded the hypothesis of dividing energy into increments as a mathematical artifice, introduced merely to get the correct answer, other physicists including Albert Einstein built on his work, and Planck's insight is now recognized to be of fundamental importance to quantum theory.

Definition

Every physical body spontaneously and continuously emits electromagnetic radiation and the spectral radiance of a body, Bν, describes the spectral emissive power per unit area, per unit solid angle and per unit frequency for particular radiation frequencies. The relationship given by Planck's radiation law, given below, shows that with increasing temperature, the total radiated energy of a body increases and the peak of the emitted spectrum shifts to shorter wavelengths. According to Planck's distribution law, the spectral energy density (energy per unit volume per unit frequency) at given temperature is given by:

u

ν

(

ν

,

T

)

=

8

π

h

ν

3

c

3

1

exp

⁡

(

h

ν

k

B

T

)

−

1

.

{\displaystyle u_{\nu }(\nu ,T)={\frac {8\pi h\nu ^{3}}{c^{3}}}{\frac {1}{\exp \left({\frac {h\nu }{k_{\mathrm {B} }T}}\right)-1}}.}

Alternatively, the law can be expressed for the spectral radiance of a body for frequency ν at absolute temperature T

given as:

B

ν

(

ν

,

T

)

=

2

h

ν

3

c

2

1

exp

⁡

(

h

ν

k

B

T

)

−

1

{\displaystyle B_{\nu }(\nu ,T)={\frac {2h\nu ^{3}}{c^{2}}}{\frac {1}{\exp \left({\frac {h\nu }{k_{\mathrm {B} }T}}\right)-1}}}

where kB is the Boltzmann constant, h is the Planck constant, and c is the speed of light in the medium, whether material or vacuum.

The spectral radiance can also be expressed per unit wavelength

λ

{\displaystyle \lambda }

instead of per unit frequency:

B

λ

(

λ

,

T

)

=

2

h

c

2

λ

5

1

exp

⁡

(

h

c

λ

k

B

T

)

−

1

{\displaystyle B_{\lambda }(\lambda ,T)={\frac {2hc^{2}}{\lambda ^{5}}}{\frac {1}{\exp \left({\frac {hc}{\lambda k_{\mathrm {B} }T}}\right)-1}}}

However this form is not related to

B

ν

(

ν

,

T

)

{\displaystyle B_{\nu }(\nu ,T)}

by the substitution

λ

=

c

/

ν

{\displaystyle \lambda =c/\nu }

. These are different functions because the spectral radiance is defined in terms of equal amounts of radiation for each increment in the independent variable (

ν

{\displaystyle \nu }

or

λ

{\displaystyle \lambda }

) and these increments are not the same in the two forms.