The graph, shown in the diagram, represents the variation of temperature (T) of the bodies, x and y

Q: Why does a steeper curve mean a higher rate of cooling?

The slope of a Temperature-Time graph represents $dT/dt$, which is the rate of change of temperature. A steeper downward curve means the temperature is dropping by a larger amount in the same time interval, hence a higher rate of cooling.

Q: How do we know the surface areas are the same?

The problem statement explicitly mentions that bodies $x$ and $y$ have the 'same surface area'. This allows us to directly compare their emissivities based solely on their cooling rates.

Q: What exactly is Kirchhoff's Law?

Kirchhoff's Law of Thermal Radiation states that for an arbitrary body in thermal equilibrium with its environment, its emissivity is equal to its absorptivity. In simpler terms, good absorbers of a particular wavelength are also good emitters of that same wavelength.

Q: Could the mass or specific heat of the bodies affect the cooling curve?

Yes, absolutely! The rate of cooling depends on $ms$ (mass $\times$ specific heat). However, in problems like this where only emissivity and absorptivity are asked to be compared, it is implicitly assumed that their thermal capacities are identical, or the question is purely testing the relation between $E$ and $a$.

Q: Why is the slope negative?

The slope is negative because the temperature is decreasing as time increases. When we talk about the 'rate of cooling', we usually refer to the magnitude of this negative slope, i.e., $|dT/dt|$.

Question

The graph, shown in the diagram, represents the variation of temperature ($T$) of the bodies, $x$ and $y$ having same surface area, with time ($t$) due to the emission of radiation. Find the correct relation between the emissivity and absorptivity power of the two bodies Options: (A) $E_x > E_y$ and $a_x < a_y$ (B) $E_x < E_y$ and $a_x > a_y$ (C) $E_x > E_y$ and $a_x > a_y$ (D) $E_x < E_y$ and $a_x < a_y$

Sigma Percentile · Accepted Answer

### Analyzing the Cooling Curves

Imagine you are observing two hot bodies, $x$ and $y$, placed in a room. Both start at the exact same high temperature, but as time passes, they cool down differently. The graph provided in the problem captures this exact scenario, plotting temperature $T$ against time $t$.

When we look at the curves, the most striking feature is their steepness. Curve $x$ dives down much more sharply than curve $y$. In the language of calculus, the magnitude of the slope of curve $x$ is greater than that of curve $y$.

Mathematically, this means the rate of cooling for body $x$ is higher:

$$ \left|-\frac{dT}{dt}ight|_x > \left|-\frac{dT}{dt}ight|_y $$

### The Role of Emissivity

Why would one body cool faster than another if they have the same surface area and are in the same environment? The answer lies in the **Stefan-Boltzmann Law**. This law tells us that the rate at which a body loses heat (and thus cools down) is directly proportional to its emissivity, $E$.

$$ 	ext{Rate of cooling} \propto E $$

Since body $x$ is cooling at a faster rate, it must be emitting thermal radiation more efficiently. Therefore, we can confidently conclude that the emissivity of body $x$ is greater than that of body $y$:

$$ E_x > E_y $$

### Kirchhoff's Law and Absorptivity

Now, how does this relate to absorptivity, $a$? To connect the two, we bring in **Kirchhoff's Law of Thermal Radiation**. It elegantly states a profound truth about nature: *good emitters are also good absorbers*.

If a body is highly efficient at radiating energy away, it is equally efficient at absorbing energy that falls upon it. This means emissivity $E$ is directly proportional to absorptivity $a$.

Since we already established that $E_x > E_y$, it naturally follows that body $x$ must also be a better absorber than body $y$:

$$ a_x > a_y $$

### The Final Verdict

Combining our findings, we see that body $x$ outperforms body $y$ in both emitting and absorbing radiation.

**Final Result:** $E_x > E_y$ and $a_x > a_y$.

This perfectly matches option (c). Always remember this visual intuition: a steeper cooling curve always points to a higher emissivity and, consequently, a higher absorptivity.

Select Answer:

Visualized Solution

Analyzing the Cooling Curves

Comparing the Rates of Cooling

Stefan-Boltzmann Law

Comparing Emissivities

Kirchhoff's Law of Radiation

Final Conclusion

The Way Forward

The Sigma Insight: Heat Transfer

Analyzing the Cooling Curves

The Role of Emissivity

Kirchhoff's Law and Absorptivity

The Final Verdict

Similar Questions

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

A body with area $A$ and temperature $T$ and emissivity $e = 0.6$ is kept inside a spherical black body. What will be the maximum energy radiated?

Two spheres of the same material have radii 1 m and 4 m and temperatures 4000 K and 2000 K, respectively. The ratio of the energy radiated per second by the first sphere to that by the second is

Variation of radiant energy emitted by sun, filament of tungsten lamp and welding arc as a function of its wavelength is shown in figure. Which of the following option is the correct match?

Select Answer:

Visualized Solution

Analyzing the Cooling Curves

Comparing the Rates of Cooling

Stefan-Boltzmann Law

Comparing Emissivities

Kirchhoff's Law of Radiation

Final Conclusion

The Way Forward

The Sigma Insight: Heat Transfer

Analyzing the Cooling Curves

The Role of Emissivity

Kirchhoff's Law and Absorptivity

The Final Verdict

Similar Questions

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A body with area A and temperature T and emissivity e=0.6 is kept inside a spherical black body. What will be the maximum energy radiated?

Two spheres of the same material have radii 1 m and 4 m and temperatures 4000 K and 2000 K, respectively. The ratio of the energy radiated per second by the first sphere to that by the second is

Variation of radiant energy emitted by sun, filament of tungsten lamp and welding arc as a function of its wavelength is shown in figure. Which of the following option is the correct match?

A body with area $A$ and temperature $T$ and emissivity $e = 0.6$ is kept inside a spherical black body. What will be the maximum energy radiated?