Question

If a star \(A\) has radiant power three times that of the Sun and a temperature of \(6000\,K\) while the Sun has temperature \(2000\,K\), what is the ratio of their radii \((R_A : R_{Sun})\)?

• A. \(1:\sqrt{27}\)
• B. \(\sqrt{27}:1\)
• C. \(1:9\)
• D. \(1:27\)

Hint:

From Stefan–Boltzmann law \(P \propto R^2T^4\). After solving for \(R^2\), always take the square root to obtain the ratio of radii.

Answer 1

The Correct Option is A

Topic - Physics: Thermal Properties of Matter / Radiation
Step 1: Understanding the Question:
We need to find the ratio of the radius of star A to the radius of the Sun based on their radiant powers and surface temperatures.
Step 2: Key Formula or Approach:
According to the Stefan-Boltzmann Law, the radiant power (\(P\)) of a blackbody is given by:
\[ P = \sigma A T^4 \]
Where \(A\) is the surface area (\(4\pi R^2\)).
Thus, \(P = \sigma (4\pi R^2) T^4\), which implies:
\[ P \propto R^2 T^4 \]
Step 3: Detailed Solution:
Let \(P_A, R_A, T_A\) be for star A and \(P_S, R_S, T_S\) be for the Sun.
Given:
\(P_A = 3 P_S \Rightarrow \frac{P_A}{P_S} = 3\)
\(T_A = 6000\,K, T_S = 2000\,K \Rightarrow \frac{T_A}{T_S} = 3\)
Using the proportionality:
\[ \frac{P_A}{P_S} = \left(\frac{R_A}{R_S}\right)^2 \times \left(\frac{T_A}{T_S}\right)^4 \]
Substitute the values:
\[ 3 = \left(\frac{R_A}{R_S}\right)^2 \times (3)^4 \]
\[ 3 = \left(\frac{R_A}{R_S}\right)^2 \times 81 \]
\[ \left(\frac{R_A}{R_S}\right)^2 = \frac{3}{81} = \frac{1}{27} \]
\[ \frac{R_A}{R_S} = \sqrt{\frac{1}{27}} \]
\[ \frac{R_A}{R_S} = \frac{1}{\sqrt{27}} \]
Step 4: Final Answer:
The ratio of radii \(R_A : R_{Sun}\) is \(1:\sqrt{27}\).