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Step-by-Step Solution
Step 1: Identify the Relevant Law
The radiant energy emitted by a body (such as the Sun), assuming it behaves like a black body, is given by the Stefan–Boltzmann law:
$ E = \sigma A T^4 $
Here,
$E$ is the total energy radiated per unit time.
$\sigma$ is the Stefan–Boltzmann constant (a universal constant).
$A$ is the surface area of the emitting body.
$T$ is the absolute temperature of the body.
Step 2: Express the Surface Area in Terms of Radius
The surface area of a sphere of radius $R$ is proportional to $R^2$. Since $A$ (for a sphere) is $4\pi R^2$, we focus on the dependence:
$ A \propto R^2 $
Thus, the energy emitted can be rewritten (up to a constant factor) as:
$ E \propto R^2 T^4 $
Step 3: Write the Ratio for the Changed Conditions
Initially, the Sun has radius $R_1 = R$ and temperature $T_1 = T$. After changes, the Sun has radius $R_2 = 2R$ and temperature $T_2 = 2T$. Hence, the ratio of the new radiant energy $E_2$ to the original radiant energy $E_1$ is:
$ \frac{E_2}{E_1} = \frac{R_2^2 \, T_2^4}{R_1^2 \, T_1^4}. $
Step 4: Substitute the Given Values
Substitute $R_2 = 2R$ and $T_2 = 2T$ into the ratio:
$ \frac{E_2}{E_1}
= \frac{(2R)^2 \,(2T)^4}{R^2 \,T^4}
= \frac{4R^2 \times 16T^4}{R^2 \,T^4}
= 64.
$
Step 5: Conclude the Result
The new radiant energy received on Earth (which is proportional to the total emission of the Sun) becomes 64 times greater than it was originally.
Answer
64
