# 26. Class 11th Physics | Heat Transfer | Wien’s Black Body Radiation Laws | by Ashish Arora

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.