Animated Solution for Physics - Properties of Solids and Liquids: A black body of temperature $T$ is inside a chamber of temperature $T_0$. Now the closed chamber is slightly opened to sun such that temperature of black body ($T$) and chamber ($T_0$) remains constant

Have you ever encountered a physics problem that seems to give you too much information? A problem that paints a vivid picture, only to test a single, fundamental principle? This question from JEE Advanced 2006 is a classic example of such a pedagogical masterpiece. It invites us to imagine a black body at a temperature $T$, resting inside a chamber at temperature $T_0$. Then, it introduces a dramatic twist: the chamber is opened to the blazing sun!

Our intuition immediately starts firing. We think about the intense solar radiation pouring into the chamber. We imagine the black body absorbing this new, massive influx of energy. We might even start mentally calculating solid angles and radiation fluxes. But before we get lost in the mathematical weeds, the problem drops a crucial, non-negotiable constraint: the temperature of the black body ($T$) and the chamber ($T_0$) remains absolutely constant.

This single constraint is the key that unlocks the entire problem. It is a gentle reminder from the examiners to step back from the complex scenario and return to the bedrock principles of thermodynamics.

To truly appreciate the elegance of this problem, we must first revisit Prevost's Theory of Heat Exchange. Proposed by Pierre Prévost in 1791, this theory revolutionized our understanding of thermal radiation. Before Prévost, people believed that cold was a substance that could be transferred. Prévost clarified that all bodies, regardless of their temperature (as long as it is above absolute zero), continuously radiate thermal energy into their surroundings.

Simultaneously, these bodies are constantly absorbing thermal energy radiated by their surroundings. The temperature of a body is simply a reflection of the balance between these two continuous processes.

If a body absorbs more energy than it emits, its internal energy increases, and its temperature rises.

Conversely, if a body emits more energy than it absorbs, its internal energy decreases, and its temperature falls.

Thermal Equilibrium

This brings us to the concept of thermal equilibrium. When a body's temperature is constant, it means its internal energy is neither increasing nor decreasing. According to the First Law of Thermodynamics, if no work is being done, the net heat exchange must be zero.

In the context of radiation, this means that the rate at which the body absorbs radiant energy must be exactly equal to the rate at which it emits radiant energy.

This is a universal truth for any body at a constant temperature. It does not matter if the body is in a dark room, inside a furnace, or exposed to the sun. If its temperature is not changing, it is perfectly balancing its energy ledger.

Now, let's apply this unshakeable principle to our specific problem. The chamber is opened to the sun. Our instinct tells us that the black body will now absorb more radiation.

Is this true? It might be! By opening the chamber, we have replaced a section of the wall (which was radiating at a relatively low temperature $T_0$) with the sun (which radiates at a massive temperature of around $5800\text{ K}$). It is highly likely that the total radiation incident on the black body has increased.

However, the problem explicitly states that the temperature $T$ remains constant.

How is this possible? Perhaps the chamber was moved further away from the sun. Perhaps the opening is incredibly small. The beauty of the problem is that we don't need to know. The mechanics of how the temperature is kept constant are irrelevant. The fact that it is constant is all that matters.

If the temperature $T$ is constant, then according to Stefan-Boltzmann's Law, the energy emitted by the black body is also constant.

Since the emission is constant, and the temperature is constant, the absorption must also be constant and equal to the emission.

Let's evaluate the given options through the lens of this core principle.

Option (a) suggests the black body will absorb more radiation. While this might seem intuitively true due to the sun's presence, we cannot definitively prove it without knowing the exact geometry of the hole and the previous state. More importantly, if it did absorb more radiation, it would have to emit more radiation to keep $T$ constant, which contradicts Stefan's Law (since $T$ is constant, emission is constant). Thus, absorption must actually be the same as before.

Option (b) suggests it will absorb less radiation. Again, we cannot prove this, and it is highly unlikely given the introduction of the sun.

Option (c) suggests the black body will emit more energy. This is demonstrably false. The energy emitted by a black body depends solely on its temperature ($E=\sigma A{T}^4$). Since $T$ is constant, the emitted energy is strictly constant.

Option (d) states that the black body emits energy equal to the energy absorbed by it. This is the definition of thermal equilibrium. Because the temperature is constant, the net heat exchange is zero. The energy going in must exactly equal the energy going out.

Therefore, option (d) is the only statement that is universally and fundamentally true given the constraints of the problem.

This question is a brilliant exercise in filtering out noise. It teaches us that in physics, a single, powerful constraint (like constant temperature) can override a complex, distracting setup (like opening a chamber to the sun). Always trust the fundamental laws!