Physics Q96 – AP-EAMCET 2025

Home Physics Thermal PropertiesQuestion #489

A solid sphere at a temperature of

400 K

radiates a power

P

. If the radius of the sphere is halved and its absolute temperature is doubled, then the power radiated by it is

Clue 1 of 3: Stefan-Boltzmann Law

The power radiated by a body is directly proportional to its surface area and the fourth power of its absolute temperature.

🧠 Full Solution Path

Step 1: Identify the governing principle and initial conditions✦ Active

The problem describes thermal radiation from a solid sphere. The power radiated by a body is given by the Stefan-Boltzmann Law, which states that the power $P$ is proportional to the surface area $A$ and the fourth power of the absolute temperature $T$ . For a sphere, the surface area is $A = 4 π r^{2}$ , where $r$ is the radius.

Therefore, the power radiated can be expressed as:

P = e σ A T^{4} = e σ (4 π r^{2}) T^{4}

Where $e$ is the emissivity and $σ$ is the Stefan-Boltzmann constant. For a given sphere, $e$ , $σ$ , and $4 π$ are constants. So, we can write the proportionality:

P \propto r^{2} T^{4}

Let the initial power be $P_{1} = P$ , initial radius be $r_{1} = r$ , and initial temperature be $T_{1} = T$ (we can use $T$ instead of $400 K$ for proportionality calculations).

💡 Teacher's Secret Hint

Remember that the Stefan-Boltzmann Law applies to absolute temperature, so always use Kelvin (K) for temperature values in these calculations.

Step 2: Define the new conditions○ Expand

According to the problem statement, the radius of the sphere is halved, and its absolute temperature is doubled.

New radius: $r_{2} = \frac{r_{1}}{2} = \frac{r}{2}$

New temperature: $T_{2} = 2 T_{1} = 2 T$

We need to find the new power radiated, $P_{2}$ .

Step 3: Set up a ratio of the power radiated under new and initial conditions○ Expand

Using the proportionality $P \propto r^{2} T^{4}$ , we can write the ratio of the new power to the initial power:

\frac{P_{2}}{P_{1}} = \frac{r_{2}^{2} T_{2}^{4}}{r_{1}^{2} T_{1}^{4}}

💡 Teacher's Secret Hint

Using ratios is often the most efficient way to solve problems involving proportional relationships when initial and final states are given.

Step 4: Substitute the new conditions and calculate the new power○ Expand

Substitute the expressions for $r_{2}$ and $T_{2}$ in terms of $r_{1}$ and $T_{1}$ (or $r$ and $T$ ) into the ratio:

\frac{P_{2}}{P} = \frac{(\frac{r}{2})^{2} (2 T)^{4}}{r^{2} T^{4}}

Simplify the expression:

\frac{P_{2}}{P} = \frac{\frac{r^{2}}{4} (16 T^{4})}{r^{2} T^{4}}

Cancel out $r^{2}$ and $T^{4}$ :

\frac{P_{2}}{P} = \frac{1}{4} \times 16

\frac{P_{2}}{P} = 4

Therefore, the new power radiated is:

P_{2} = 4 P

💡 Teacher's Secret Hint

Be careful with exponents, especially when squaring a fraction or raising a product to a power, e.g., $(2 T)^{4} = 2^{4} T^{4} = 16 T^{4}$ .

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Physics Question 96 – AP-EAMCET 2025

Clue 1 of 3: Stefan-Boltzmann Law

🧠 Full Solution Path