Animated Solution for Physics - Properties of Solids and Liquids: A spherical black body with a radius of 12 cm radiates 450 W power at 500 K. If the radius were halved and the temperature doubled, the power radiated in watt would be

Imagine you are floating in the vast, freezing emptiness of space, observing a perfectly spherical black body. It is glowing, radiating heat into the void. The physics governing this glowing sphere is beautifully captured by one of the most elegant laws in thermodynamics: the Stefan-Boltzmann Law.

In this problem, we are given an initial state: a sphere with a radius $R=12\text{ cm}$, a temperature $T=500\text{ K}$, and it radiates a power $P=450\text{ W}$. Then, the universe plays a trick on us. The sphere shrinks to half its radius, but its temperature doubles. Our mission is to find the new power it radiates.

The Stefan-Boltzmann Law tells us that the power $P$ radiated by a black body is directly proportional to its surface area $A$ and the fourth power of its absolute temperature $T$. Mathematically, this is written as:

Here, $\sigma$ is the Stefan-Boltzmann constant. Since our object is a sphere, its surface area is given by the geometric formula $A=4\pi R^2$. Substituting this into our power equation, we get:

Because $\sigma$ and $4\pi$ are constants, we can strip away the clutter and focus on the variables that actually change. The radiated power is proportional to the square of the radius and the fourth power of the temperature:

When dealing with "before and after" scenarios in physics, setting up a ratio is the most powerful tool in your arsenal. It allows you to bypass messy constants and unit conversions. Let's define our new state with primes: $P^{\prime }$, $R^{\prime }$, and $T^{\prime }$. We can write the ratio of the new power to the old power as:

The problem states that the new radius is half the original ($R^{\prime }=\frac{R}{2}$) and the new temperature is double the original ($T^{\prime }=2T$). This means our ratios are simply $\frac{R^{\prime }}{R}=\frac{1}{2}$ and $\frac{T^{\prime }}{T}=2$. Let's substitute these into our master equation:

Now, we just need to execute the algebra. Squaring one-half gives us one-fourth, and raising two to the fourth power gives us sixteen:

This tells us a fascinating physical truth: even though the sphere shrank, the massive increase in temperature (which is raised to the fourth power!) completely overpowered the loss of surface area. The new sphere radiates four times as much power as the original one.

To find the final numerical answer, we simply multiply this factor by the initial power:

And there we have it! The new power radiated is $1800\text{ W}$.