Wien's Displacement Law

Chandu is a very naughty boy.

Chandu's father is an eminent scientist.

One day Chandu visited the laboratory of his father where he enjoyed with the thermometer.

Chandu measured the temperature of the water which is recently taken from the flame.

He measured the temperature of the water at certain intervals of time.

Chandu found that the temperature is decreasing every time.

Since the water pot is open to the atmosphere, so there is continuous heat transfer from the water to the atmosphere.

Thus, in every measurement, the temperature is decreasing.

Let's find the temperature after an interval.

Consider a body cools down from $60 ° C$ to $40 ° C$ in $7 m i n$ .

We have to find the temperature of the body after the next $7 m i n$ . The atmospheric temperature is $10 ° C$ .

Let's generalise the case for the given conditions,

By modified Newton's law,

Modified Newton's law after time $t_{1}$ ,

Modified Newton's law after time $t_{2}$ ,

Subtracting the equations obtained at $t_{2}$ from the equation obtained at $t_{2}$ .

Further simplifying the subtraction to obtain the equation of variation of temperature of the body with time.

The equation of variation of the temperature of the body with time is as,

Putting $T_{1} = 70 ° C$ , $T_{2} = 40 ° C$ $T_{0} = 10 ° C$ and $t_{1} = 7 min$ in the equation of variation of temperature of a body.

Putting $T_{1} = 40 ° C$ , $T_{0} = 10 ° C$ and $t_{2} = 7 min$ in the equation of variation of temperature of a body.

The right-hand side of both the equations obtained from Newton's modified law is equal, thus, their left hand side should also be equal.

Equating the equations obtained from Newton's law to obtain the temperature $T_{2}$

Further proceeding to find the $T_{2}$ .

The temperature of the body is changed due to the heat transfer from the body to the surrounding.

Heat transfer between two bodies can take place in three modes; conduction, convection and radiation.

Wein has given the law for the radiation heat transfer known as Wein's displacement law.

Let's study the Wein's displacement law for the radiation heat transfer.

The Wien's Displacement Law state that the wavelength carrying the maximum energy is inversely proportional to the absolute temperature of a black body.

It is known very clear, everybody above absolute zero temperature (0 K) emits the thermal radiation.

A body at a temperature radiate energy in the form of waves. The wavelength of all the waves is not the same, but the waves emitted will form a spectrum of wavelengths.

The wavelength at a temperature corresponding to which maximum energy is emitted is maximum wavelength.

Therefore, Wein's displacement law is also written, the product of temperature and maximum wavelength occurring at that temperature.

Therefore, Wein's displacement law is as,

If the graph is drawn between the Energy emitted in the $y - a x i s$ and the wavelength in the $x - a x i s$ at a temperature.

The wavelength at which the maximum energy is emitted will be the peak point in the curve of temperature says $T_{1}$ .

If the temperature is changed to $T_{2}$ and $T_{2} > T_{1}$ , the maximum energy will be emitted at shorter wavelength.

The $λ_{1} > λ_{2}$ , as per Wein's displacement law because $T_{2} > T_{1}$ .

'And' hence, it can be concluded that as the temperature increases the maximum wavelength becomes shorter but the area inside the curve increases.

The Wein's displacement law in terms of frequency can be written by replacing wavelength.

Revision

The Wien's Displacement Law state that the wavelength carrying the maximum energy is inversely proportional to the absolute temperature of a black body.

Everybody above absolute zero temperature (0 K) emits the thermal radiation.

It can be concluded that as the temperature increases the maximum wavelength becomes shorter but the area inside the curve increases.

The End