another useful discovery, related to black
body radiation was done by, wien’s experiments. wien deduced, various laws to understand,
black body radiation, in terms of, its wavelength. we’ve studied that, when a body is heated,
it starts emitting, an electromagnetic radiation, to the surrounding which we call, thermal
radiation. and these electromagnetic radiations consists of several wavelengths. so, when
the black body radiation, when a black body is heated and radiations are emitted, these
are analyzed, by using a spectrometer, that is, the energy contained in specific wavelengths,
of the radiation emitted, are analyzed, several facts came out, let’s discuss those. this
is a black body which is heated, and a graph was plotted, for, the wavelength and the energy
emitted corresponding to individual wavelengths. which can be given as spectral emission energy
per unit wavelength range. in this situation when the graph is plotted, the graph looks
like this, for a given temperature of the body. say the body temperature was thousand
kelvin. at this temperature, the graph was plotted, then it was observed that at a particular
wavelength, lambda m 1, the graph was getting a peak. that means the energy corresponding
to this wavelength from the black body will be maximum. and when temperature was increased,
and again experimentally the graph was plotted for energy corresponding to various wavelengths.
the graph was obtained like this. say it is obtained at 12 hundred kelvin temperature
for the black body. it was seen, that the peak, was obtained at a higher temperature
corresponding to slightly lower wavelength, say it is lambda m 2. and again if the temperature
is increased, a similar observation was recorded, say at higher temperature 15 hundred kelvin.
again the peak was shifted to a lower wavelength, lambda m 3 was further lower. so this was
analyzed by wien, and summarized in form of, a law, which we call wien’s displacement
law. and, it was stated in a way that, it was observed. that. as temperature of black
body increases. the wavelength. corresponding to. the peak. radiation intensity. shifts
towards. lower wavelengths. and , this implies that, wavelength corresponding to peak radiation
intensity, is decreasing with rise in temperature, and mathematically it was, analyzed by wien
that this lambda m is inversely proportional to t, there’s an inverse proportionality
relation between, the wavelengths corresponding to peak emission of energy from a black body,
and its temperature. so, a proportionality constant was included, to make it an equality
which is, b by t. this is the relation developed by wien, where this b was termed as, wien’s
constant. and numerical value of, b is, 2 point 8 9 into, 10 to power minus 3, meter
kelvin. this is the value which was, analyzed by wien. and this is the expression which
was termed as, wien’s displacement law. this is quite a useful law when we analyze
the spectral intensity of radiations, emitted from a black body. and based on this, several
various, other laws were also derived. another 1 was analyzed by rayleigh, under rayleigh
analysis, that also we’ll see. for spectral energy distribution of a black
body, rayleigh developed a rough relationship, called rayleigh analysis. we can write about
it that, it was analyzed. roughly. that for higher wavelengths. spectral intensity. spectral
intensity is the energy emitted per unit area, from a black body, per unit range of wavelengths.
this spectral intensity is given as, this was denoted as e lambda, it was obtained as,
2 pi, c k t, upon lambda to power 4. always, keep in mind that this expression was developed
by rayleigh for, higher wavelengths, and it is not valid. for lower wavelengths. that
means it is partially correct. we cannot use this relation for, all wavelengths. so we
can write, at maximum, spectral intensity, when we were talking about peak of, wien’s
black body radiation spectrum, lambda will be lambda m, and we can write e lambda will
be equal to, e lambda maximum, that is peak of the curve. so by using the relation we
can write, e lambda maximum will be equal to, 2 pi, c k t, divided by lambda m to power
4. and, as we’ve already studied, by wien’s displacement law, we can use lambda m as,
b by t, so we substitute it here. this will give us e lambda max, is equal to, 2 pi, c
k t, divided by, b upon t to power 4. so, this’ll be, 2 pi, c k t to power 5 upon,
b to power 4. so here we can write, in the black body radiation spectrum, the peak spectral
intensity emitted from the black body will be directly proportional to, t to power 5.
this is another important relation, for black body radiation spectrum. and this is, wien’s,
5th power law.

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