Newton’s Law of Cooling: Class 11 Physics Notes, Definition, Formula, Derivation, and Application

Put simply, according to Newton's Law of Cooling, the hotter an object is as compared to its surroundings, the faster it cools down.

This principle applies to several scenarios where heat needs to be managed. Think of a CPU that needs a cooling fan to prevent overheating. Or, consider the car's engine. Manufacturers design radiators and cooling systems so that the engine does not burn after a long drive. All these are based on this law of Newton. 

Considering these examples, Newton's Law of Cooling helps calculate how fast or slow something cools down. The precise mathematical formula for it is 

$\frac{\mathrm{dT}}{\mathrm{dt}} = - k ( T - T_{\mathrm{env}} )$

$$\begin{array}{lll}T\left(t\right)& =& \text{temperature at time}t\\ T_0& =& \text{initial temperature}\\ T_{\text{env}}& =& \text{surrounding temperature}\\ k& =& \text{cooling constant}\\ e& =& \text{mathematical constant (≈ 2.718)}\end{array}$$

 

Importance of Newton's Law of Cooling for Engineering Exams

• Previous years' JEE Mains questions on Newton's Law of Cooling are around solving numerical problems, interpreting graphs, explaining application-based scenarios, or related concepts like Stephan Boltzmann Law. 
• Newton's Law of Cooling is a gateway concept to understand cooling of electronics, perform basic thermal physics experiments, and for specialising across different venues of engineering (mechanical and aerospace engineering).  

Section 10.10 of the Thermal Properties of Matter chapter, mentions -  

"According to Newton’s law of cooling, the rate of loss of heat, – dQ/dt of the body is directly proportional to the difference of temperature ∆T = (T2–T1) of the body and the surroundings."

Also, this principle applies when certain conditions meet.  

• Temperature difference between the object and the surrounding has to be small. 
• The nature of the surface will determine the cooling. For instance, dark surfaces radiate heat faster. 
• If the surface area is large, the cooling will be faster. 
• The airflow of the medium will affect the cooling rate.  

For small temperature difference between a body and its surrounding, the rate of cooling of the body is directly proportional to the temperature difference and the surface area exposed.

$\frac{d\theta }{dt}\propto (\theta –{\theta }_0),\mathrm{}\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}\mathrm{}\mathrm{}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{}{\theta }_0$ are temperature corresponding to object and surroundings. 

 

From above expression, $\frac{d\theta }{dt}=-k(\theta -{\theta }_0)$ .....(14.1)

This expression represents Newton's law of cooling.

Let's see how to derive Newton's Law of Cooling from Stefan's Law.

$k=\frac{4e\sigma {\theta }_0^3}{mc}$ A .....(14.2)

 

Sometime when we need only approximate values from Newton’s law, we can assume a constant rate of cooling, which is equal to the rate of cooling corresponding to the average temperature of the body during the interval.

$\frac{d\theta }{dt}=-k\left[\theta -{\theta }_0\right]\underset{{\theta }_i}{\overset{{\theta }_f}{\int }}\frac{d\theta }{\left(\theta -{\theta }_0\right)}=\underset{0}{\overset{t}{\int }}-kdt$ .....(14.4)

 

 

Go through these notes, aligned with the latest NCERT syllabus. 

 

Units and Measurements Class 11 Notes	Mechanical Properties of Solids Class 11 Notes
Motion in a Straight Line Class 11 Notes	Mechanical Properties of Fluids Class 11 Notes
NCERT Class 11 Notes for Motion in a Plane	Thermal Properties of Matter Class 11 Notes
Laws of Motion Class 11 Notes	Thermodynamics Class 11 Notes
Work, Energy, and Power Class 11 Notes	Kinetic Theory of Gas Class 11 Notes
System of Particles and Rotational Motion Class 11 Notes	Oscillations Class 11 Notes
Gravitation Class 11 Notes	Waves Class 11 Notes

 

NCERT Class 11 Notes for PCM

NCERT Class 11 Physics Notes

 

Explore NCERT Solutions for Physics Class 11. Fond more below.  

Units and Measurements Class 11 NCERT Solutions	Mechanical Properties of Solids Class 11 NCERT Solutions
Motion in a Straight Line Class 11 NCERT Solutions	Mechanical Properties of Fluids Class 11 NCERT Solutions
Motion in a Plane Class 11 NCERT Solutions	Thermal Properties of Matter Class 11 NCERT Solutions
Laws of Motion Class 11 NCERT Solutions	Thermodynamics Class 11 NCERT Solutions
Work, Energy, and Power Class 11 NCERT Solutions	Kinetic Theory Class 11 NCERT Solutions
System of Particles and Rotational Motion Class 11 NCERT Solutions	Oscillations Class 11 NCERT Solutions
Gravitation Class 11 NCERT Solutions	Waves Class 11 NCERT Solutions