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Newton's Law of Cooling Notes | Study Physics For JEE - JEE

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8. Newton's law of cooling

According to this law, if the temperature T of the body is not very different from that of the surroundings T0, then rate of cooling - Newton`s Law of Cooling Notes | Study Physics For JEE - JEE is proportional to the temperature difference between them. To prove it let us assume that

T = T0 + Dt

Newton`s Law of Cooling Notes | Study Physics For JEE - JEE

Newton`s Law of Cooling Notes | Study Physics For JEE - JEENewton`s Law of Cooling Notes | Study Physics For JEE - JEE

if the temperature difference is small.

Thus, rate of cooling

Newton`s Law of Cooling Notes | Study Physics For JEE - JEE or Newton`s Law of Cooling Notes | Study Physics For JEE - JEE

as dT = dq or DT = Dq

8.1 Variation of temperature of a body according to Newton's law 

Suppose a body has a temperature qi at time t = 0. It is placed in an atmosphere whose temperature is q0. We are interested in finding the temperature of the body at time t, assuming Newton's law of cooling to hold good or by assuming that the temperature difference is small. As per this law,

Newton`s Law of Cooling Notes | Study Physics For JEE - JEENewton`s Law of Cooling Notes | Study Physics For JEE - JEE

rate of cooling µ temperature difference

or Newton`s Law of Cooling Notes | Study Physics For JEE - JEE or Newton`s Law of Cooling Notes | Study Physics For JEE - JEE

Here Newton`s Law of Cooling Notes | Study Physics For JEE - JEE is a constant

Newton`s Law of Cooling Notes | Study Physics For JEE - JEE

Newton`s Law of Cooling Notes | Study Physics For JEE - JEE

From this expression we see that q = qi at t = 0 and q = qat t = ¥, i.e., temperature of the body varies exponentially with time from qi to q0 (< qi). The temperature versus time graph is a shown in figure.

Newton`s Law of Cooling Notes | Study Physics For JEE - JEE

Note : If the body cools by radiation from q1 to q2 in time t, then taking the approximation

Newton`s Law of Cooling Notes | Study Physics For JEE - JEE and Newton`s Law of Cooling Notes | Study Physics For JEE - JEE

The equation Newton`s Law of Cooling Notes | Study Physics For JEE - JEE    becomes

Newton`s Law of Cooling Notes | Study Physics For JEE - JEE

This form of the law helps in solving numerical problems related to Newton's law of cooling. 

8.2 Limitations of Newton's Law of Cooling : 

(a) The difference in temperature between the body and surroundings must be small

(b) The loss of heat from the body should be radiation only.

(c) The temperature of surroundings must remain constant during the cooling of the body.

Ex.20 A body at temperature 40°C is kept in a surrounding of constant temperature 20°C. It is observed that its temperature falls to 35°C in 10 minutes. Find how much more time will it take for the body to attain a temperature of 30°C. 

Sol.

Newton`s Law of Cooling Notes | Study Physics For JEE - JEE

for the interval in which temperature falls from 40 to 35°C

Newton`s Law of Cooling Notes | Study Physics For JEE - JEE

for the next interval
(30 - 20) = (35 - 20) e-kt Newton`s Law of Cooling Notes | Study Physics For JEE - JEE

Newton`s Law of Cooling Notes | Study Physics For JEE - JEE
Newton`s Law of Cooling Notes | Study Physics For JEE - JEE

Alter: (by approximate method)
for the interval in which temperature falls from 40 to 35°C

Newton`s Law of Cooling Notes | Study Physics For JEE - JEE
from equation  Newton`s Law of Cooling Notes | Study Physics For JEE - JEE
Newton`s Law of Cooling Notes | Study Physics For JEE - JEE
for the interval in which temperature falls from 35°C to 30°C

Newton`s Law of Cooling Notes | Study Physics For JEE - JEE

from equation (14.4)

Newton`s Law of Cooling Notes | Study Physics For JEE - JEE

⇒ required time,
Newton`s Law of Cooling Notes | Study Physics For JEE - JEE 

9. NATURE OF THERMAL RADIATIONS : (WIEN'S DISPLACEMENT LAW) 

From the energy distribution curve of black body radiation, the following conclusions can be drawn :

(a) The higher the temperature of a body, the higher is the area under the curve i.e. more amount of energy is emitted by the body at higher temperature.

(b) The energy emitted by the body at different temperatures is not uniform. For both long and short wavelengths, the energy emitted is very small.

Newton`s Law of Cooling Notes | Study Physics For JEE - JEE

(c) For a given temperature, there is a particular wavelength (lm) for which the energy emitted (El) is maximum

(d)  With an increase in the temperature of the black body, the maxima of the curves shift towards shorter wavelengths.

From the study of energy distribution of black body radiation discussed as above, it was established experimentally that the wavelength (lm) corresponding to maximum intensity of emission decreases inversely with increase in the temperature of the black body. i.e.

Newton`s Law of Cooling Notes | Study Physics For JEE - JEE

This is called Wien's displacement law.

Here b = 0.282 cm-K, is the Wien's constant.

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