Hot water is poured simultaneously into four identical metallic tumblers that are painted on the outside with different types of surfaces. After some time, in which tumbler will the water be found to have cooled the most due to maximum heat loss by radiation?

Introduction / Context:
This question is about heat transfer by radiation and how the nature and colour of a surface affect its ability to emit and absorb thermal radiation. In everyday life, black and rough surfaces are known to get hotter in the Sun and to cool more quickly in the shade. Physics explains this behaviour by the concepts of good and poor radiators. Understanding which type of surface radiates heat most effectively helps answer questions involving cooling and heating of bodies.

Given Data / Assumptions:

• Four identical metallic tumblers contain hot water initially at the same temperature.
• The tumblers differ only in the nature and colour of their outer surfaces: rough white, rough black, shining white and shining black.
• The tumblers are kept under identical environmental conditions.
• We are asked in which tumbler the water cools the most, that is, where heat loss by radiation is maximum.

Concept / Approach:
According to thermal radiation principles, a good absorber of heat is also a good emitter, and poor absorbers are also poor emitters. Black, dull and rough surfaces are excellent absorbers and emitters, while white, shiny and polished surfaces are poor absorbers and emitters. Therefore, a rough black surface will radiate heat away more effectively than a shining or white surface. As a result, the tumbler with rough black paint will allow the hot water to lose heat faster and cool more in a given time.

Step-by-Step Solution:
Step 1: Recall that black surfaces absorb and emit thermal radiation more strongly than white surfaces, which mostly reflect radiation. Step 2: Understand that rough or matte surfaces behave as better radiators than smooth, shiny surfaces with the same colour. Step 3: Combine these ideas: a rough black surface is both black (good emitter) and rough (good radiator), making it the best radiator among the given options. Step 4: Recognise that rough white is a good radiator compared to shiny surfaces but still less effective than rough black because white reflects more radiation. Step 5: Shining white and shining black surfaces reflect more radiation and emit less, making them poorer radiators than rough surfaces. Step 6: Therefore, the water in the rough black painted tumbler will lose heat fastest and cool the most.

Verification / Alternative check:
An experimental way to verify this is to fill identical containers with hot water and paint them with different surface finishes. After some time, measuring the temperature of the water will show that the container with a dull black surface leads to the greatest temperature drop. This experiment is often demonstrated in school laboratories and supports the theoretical prediction that rough black is the best radiator. The same idea explains why solar cookers and radiators are often painted black and rough.

Why Other Options Are Wrong:
Rough white, although rough, is white and therefore reflects more radiation than black surfaces; it is a poorer radiator than rough black. Shining white is both reflective and smooth, making it one of the poorest radiators among the options. Shining black, while black, is shiny and reflects more radiation than a rough surface, so it is less effective at radiating heat than rough black.

Common Pitfalls:
Students sometimes give importance only to colour and ignore the effect of surface roughness, or vice versa. The correct understanding is that both colour and surface texture influence radiation: black is better than white, and rough is better than smooth for emission. Another confusion is between absorption and emission; however, good absorbers are also good emitters. Keeping these relationships clear helps you choose rough black as the best radiator for cooling in such questions.

Final Answer:
The water will cool the most in the tumbler painted rough black, because it is the best radiator of heat.