Radiation

Step 1: Understand radiation coefficients. When radiation falls on a body, three things may happen:
• A part is absorbed
• A part is reflected
• A part is transmitted Hence total incident energy is divided as: $ t=0 $

The radiant energy emitted by a star is given by: For spherical stars: where is the radius. Hence: Step 1: Write the energy ratio.
For two stars, the ratio of radiant energies is: Given: Substitute: Simplify: Step 2: Simplify the temperature ratio.
The ratio of temperatures is: Raise to the power of 4: Substitute back: Multiply through by 81: Step 3: Take the square root.
Thus, the ratio of their radii is:

To find the temperature of the second black body, we use the Stefan-Boltzmann Law, which states that the power radiated by a black body is given by , where is the Stefan-Boltzmann constant, is the surface area, and is the absolute temperature.

Given two black bodies radiating the same power:

Since :

Canceling out the common terms, we have:

Substitute the given values , , :

Simplifying:

Taking the fourth root:

Thus, the temperature of the second black body is .

Step 1: Understanding the concept of half-life. The half-life of a radioactive substance is the time required for half of the atoms in a sample to decay. In this case, the half-life of the substance is given as 5 years. Step 2: Probability of decay within the half-life. The probability that a single atom will decay within one half-life is 50%, since half of the atoms will have decayed by the end of one half-life. This can be understood using the fact that in each half-life, the number of undecayed atoms reduces to half of the previous amount. Therefore, in a given half-life, the probability of a single atom decaying is: Step 3: Conclusion. The probability that a single atom of the radioactive substance will decay within 5 years is . Answer: Therefore, the probability that the atom will decay within 5 years is .

Concept: The total radiant power emitted by a star is given by the Stefan–Boltzmann law: where = radiant power, = Stefan–Boltzmann constant, = surface area of the star, = absolute temperature. Substituting the surface area: Thus,

Step 1: Write the ratio of powers.
Given:

Step 2: Calculate the temperature ratio.


Step 3: Substitute into the power relation.


Step 4: Take square root to obtain radius ratio.


Step 5: Final result.