Animated Solution for Physics - Properties of Solids and Liquids: If the temperature of the sun were to increase from $T$ to $2T$ and its radius from $R$ to $2R$, then the ratio of the radiant energy received on earth to what it was previously, will be

Imagine the Sun radiating energy in all directions, and our Earth sitting millions of kilometers away, catching a tiny fraction of it. This problem asks us to determine how the energy received by Earth changes if the Sun were to undergo a massive transformation—specifically, doubling both its radius and its temperature. To solve this, we need to understand the journey of light from the surface of the star to our planet.

The Heart of the Star

First, we must determine how much total energy the Sun emits per second. According to the Stefan-Boltzmann Law, the power $P$ radiated by a black body depends on its surface area $A$ and the fourth power of its absolute temperature $T$. The formula is given by:

Since the Sun is a sphere of radius $R$, its surface area is $4\pi R^2$. Substituting this into our equation, the total power radiated by the Sun becomes:

Now, this immense energy spreads out spherically into the vacuum of space. By the time it reaches Earth at a distance $R_{SE}$, it has spread over a giant imaginary sphere of radius $R_{SE}$. The intensity $I$, which is the power per unit area at Earth's location, is the total power divided by the area of this giant sphere:

The Earth intercepts this radiation with its own cross-sectional area, $A_E$. Therefore, the total radiant energy $E$ received by Earth is simply the intensity multiplied by Earth's area:

Notice something crucial here: the distance to the Earth ($R_{SE}$) and the Earth's cross-sectional area ($A_E$) are constants. This means the energy received by Earth is directly proportional to the total power radiated by the Sun:

The problem states that the Sun's radius doubles to $2R$, and its temperature doubles to $2T$. Let's plug these new values into our proportionality relation to find the new power $P_{new}$:

Let's expand the terms carefully. Squaring the new radius gives $4R^2$. Raising the new temperature to the fourth power gives $16T^4$.

Multiplying the constants $4$ and $16$ gives $64$.

This means the new power is exactly $64$ times the original power ($P_{new}=64P_{old}$). Because the energy received by Earth is directly proportional to the Sun's total power, the Earth will receive $64$ times more radiant energy. The correct option is (d).