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Step-by-Step Solution
Step 1: Identify the Relevant Law
A spherical black body radiates power according to the Stefan-Boltzmann law, which states that the power radiated by a black body is proportional to its surface area and to the fourth power of its absolute temperature. Symbolically,
$P \propto r^2 T^4$,
where $r$ is the radius of the sphere and $T$ is its absolute temperature (in kelvins).
Step 2: Note the Given Quantities
• Initial radius: $r_1 = 12 \text{ cm}$
• Final radius: $r_2 = \frac{r_1}{2} = 6 \text{ cm}$
• Initial temperature: $T_1 = 500 \text{ K}$
• Final temperature: $T_2 = 2 \times T_1 = 1000 \text{ K}$
• Initial power radiated: $P_1 = 450 \text{ W}$
Step 3: Set Up the Proportionality Relationship
Because $P \propto r^2 T^4$, we can write the ratio of the powers as:
$$
\frac{P_2}{P_1} = \frac{r_2^2 \, T_2^4}{r_1^2 \, T_1^4}.
$$
Step 4: Substitute the Known Values
Substitute $r_1, r_2, T_1,$ and $T_2$ into the ratio:
$$
\frac{P_2}{450} = \frac{(6 \text{ cm})^2 \times (1000 \text{ K})^4}{(12 \text{ cm})^2 \times (500 \text{ K})^4}.
$$
Step 5: Simplify Step by Step
1. Radius part:
$$
\frac{(6 \text{ cm})^2}{(12 \text{ cm})^2} = \frac{36}{144} = \frac{1}{4}.
$$
2. Temperature part:
$$
\frac{(1000 \text{ K})^4}{(500 \text{ K})^4} = \left(\frac{1000}{500}\right)^4 = (2)^4 = 16.
$$
3. Combine these factors:
$$
\frac{1}{4} \times 16 = 4.
$$
Therefore,
$$
\frac{P_2}{450} = 4 \quad \Longrightarrow \quad P_2 = 450 \times 4 = 1800 \text{ W}.
$$
Step 6: State the Final Answer
Hence, when the radius is halved and the temperature is doubled, the new radiated power is
$$
P_2 = 1800 \text{ W}.
$$
