Animated Solution for Physics - Properties of Solids and Liquids: According to Newton's law of cooling, the rate of cooling of a body is proportional to $(\Delta \theta {)}^n$, where $\Delta \theta$ is the difference of the temperature of the body and the surroundings, then $n$ is equal to

A Hot Body in a Cool Room

Imagine a piping hot cup of coffee resting on your table. Over time, it cools down. But how fast does it cool? This is exactly what Newton's Law of Cooling addresses. It provides a mathematical framework to understand the everyday phenomenon of objects reaching thermal equilibrium with their environment.

Sir Isaac Newton observed that the rate at which a body loses heat to its surroundings is directly proportional to the temperature difference between the body and the surroundings, provided this difference is relatively small.

Mathematically, if $T$ is the temperature of the body and $T_s$ is the temperature of the surroundings, the temperature difference is denoted as $\Delta \theta =T-T_s$.

The rate of heat loss is given by the relation:

This means that if the temperature difference is doubled, the rate at which heat escapes the body is also doubled. It is a perfectly linear relationship.

The problem states that the rate of cooling is proportional to $(\Delta \theta {)}^n$.

By comparing this given relation with the standard statement of Newton's Law of Cooling, we can write:

This simple comparison immediately implies that the exponent $n$ must be exactly equal to $1$.

This is a fundamental theoretical principle in thermodynamics. The rate of cooling depends linearly on the temperature difference for small variations. Therefore, the correct value of $n$ is $1$, which corresponds to option (d).