A solid sphere of copper of radius R and a hollow sphere of the same material of inner radius r and

Q: Why doesn't the inner surface of the hollow sphere radiate heat?

It does radiate heat, but it radiates it into the inner cavity, where it is immediately re-absorbed by the opposite walls of the cavity. Therefore, the net heat loss from the inner surface to the outside environment is zero. Only the outer surface contributes to the net heat loss.

Q: Does the hollow sphere have a lower specific heat?

No, specific heat ($s$) is an intrinsic property of the material. Since both spheres are made of copper, their specific heat is exactly the same. What is different is their *heat capacity* ($ms$), because their masses are different.

Q: What if the hollow sphere had the same mass as the solid sphere?

If they had the same mass, the hollow sphere would need to have a much larger outer radius to compensate for the empty space inside. A larger outer radius means a larger surface area, so it would lose heat faster and still end up cooling faster!

Q: Does the thickness of the hollow sphere matter?

As long as it is hollow, its mass is less than the solid sphere. A thinner shell means even less mass, which means it has even less thermal inertia and will cool even faster.

Q: Why do we use a negative sign in the rate of cooling?

The negative sign simply indicates that the temperature is decreasing with time. Because $dT/dt$ is a negative value during cooling, adding a negative sign ($-dT/dt$) makes it a positive quantity that represents the 'speed' of cooling.

Question

A solid sphere of copper of radius $R$ and a hollow sphere of the same material of inner radius $r$ and outer radius $R$ are heated to the same temperature and allowed to cool in the same environment. Which of them starts cooling faster?

Sigma Percentile · Accepted Answer

### The Tale of Two Spheres: Heat Loss vs. Temperature Drop

Imagine two buckets of water. One is a large, wide bucket, and the other is a bucket of the exact same size, but with a large solid block placed inside it, meaning it holds much less water overall. Now, imagine both buckets have the exact same size hole at the bottom. Water will leak out of both buckets at the exact same rate. However, because the second bucket holds much less water, its water level will drop much faster than the first bucket.

This analogy perfectly illustrates the core concept of this problem: the critical difference between **heat loss** and **temperature drop**.

### Analyzing the Setup

We are given two copper spheres: one solid and one hollow. They share the same outer radius $R$ and are heated to the same initial temperature $T$.

According to the Stefan-Boltzmann Law, the rate at which a body radiates heat to its surroundings is given by:

$$\frac{dQ}{dt} = e \sigma A (T^4 - T_0^4)$$

Let's break this down. Both spheres are made of copper, so their emissivity $e$ is identical. They have the same outer radius $R$, which means their outer surface area $A = 4\pi R^2$ is exactly the same. Finally, they are at the same temperature $T$ in the same environment $T_0$.

Because all these factors are identical, both the solid and hollow spheres will lose heat energy at the exact same rate.

### The Master Equation

But losing heat is not the same as dropping in temperature. The rate of cooling (how fast the temperature drops) is related to the heat loss by the specific heat formula:

$$\frac{dQ}{dt} = -ms \frac{dT}{dt}$$

Here, $m$ is the mass, $s$ is the specific heat capacity of the material, and $-\frac{dT}{dt}$ represents the rate of cooling. If we rearrange this equation to solve for the rate of cooling, we get:

$$-\frac{dT}{dt} = \frac{\frac{dQ}{dt}}{ms}$$

### Final Calculation

We already established that the rate of heat loss $\frac{dQ}{dt}$ is the same for both spheres. Since both are made of copper, their specific heat $s$ is also the same.

This leaves us with a simple inverse relationship:

$$-\frac{dT}{dt} \propto \frac{1}{m}$$

The rate of cooling is inversely proportional to the mass. Because the hollow sphere has a cavity inside, its mass $m_{	ext{hollow}}$ is strictly less than the mass of the solid sphere $m_{	ext{solid}}$.

Less mass means less thermal inertia. It takes less heat loss to cause a significant drop in temperature. Therefore, the hollow sphere will experience a greater rate of cooling and will start cooling faster than the solid sphere.

Visualized Solution

Visualizing the Setup

Stefan-Boltzmann Law

Comparing Heat Loss

Heat Loss vs Temperature Drop

Rate of Cooling Expression

Mass Dependency

Final Conclusion

The Way Forward

The Sigma Insight: Heat Transfer

The Tale of Two Spheres

Analyzing the Setup

The Master Equation

Final Calculation

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