Assuming the sun to have a spherical outer surface of radius r, radiating like a black body at temperature $t^{\circ} C$, the power received by a unit surface, (normal to the incident rays) at distance $R$ from the centre of the Sun is:- Where $\sigma$ is the Stefan's Constant.

Question

A 200 g sample of water at 80°C is mixed with 100 g of water at 20°C. Assuming no heat loss to the surroundings, what is the final temperature of the mixture?

Answer 1

To find the power received per unit area by a surface normal to the incident rays at a distance R from the center of the Sun, we need to understand the concept of black body radiation and apply Stefan-Boltzmann Law.

Understanding Black Body Radiation: A black body is an idealized physical body that absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence. It also radiates energy at the maximum possible rate per unit area at each wavelength for any given temperature, as described by Stefan-Boltzmann Law.
Stefan-Boltzmann Law: The power P radiated by a black body per unit area is given by the formula: P = \sigma T^4, where \sigma is the Stefan's constant and T is the absolute temperature of the black body in Kelvin.
Converting Temperature: The surface temperature of the Sun is given as t^{\circ} C. To convert this to Kelvin, use: T = t + 273.
Total Power Radiated by the Sun: The Sun radiates energy over its entire surface area. Since the Sun can be approximated as a sphere with radius r, the total power P_{\text{total}} emitted by the Sun is: P_{\text{total}} = \sigma (t + 273)^4 \cdot 4 \pi r^2, where 4 \pi r^2 is the surface area of the Sun.
Intensity at Distance R: The power that reaches a unit area at distance R from the Sun is the power distributed over the surface area of a sphere with radius R. Therefore, the intensity I received is: I = \frac{P_{\text{total}}}{4 \pi R^2} = \frac{\sigma (t + 273)^4 \cdot 4 \pi r^2}{4 \pi R^2}.
Simplifying the expression gives: I = \frac{r^2 \sigma (t + 273)^4}{R^2}. This matches the first option, confirming that the correct power received by a unit surface is: r^2 \sigma (t + 273)^4 / R^2.

Thus, the correct answer is Option 1: r^2 \sigma (t + 273)^4 / R^2.

Answer 2

To find the power received per unit area by a surface normal to the incident rays at a distance R from the center of the Sun, we need to understand the concept of black body radiation and apply Stefan-Boltzmann Law.

Understanding Black Body Radiation: A black body is an idealized physical body that absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence. It also radiates energy at the maximum possible rate per unit area at each wavelength for any given temperature, as described by Stefan-Boltzmann Law.
Stefan-Boltzmann Law: The power P radiated by a black body per unit area is given by the formula: P = \sigma T^4, where \sigma is the Stefan's constant and T is the absolute temperature of the black body in Kelvin.
Converting Temperature: The surface temperature of the Sun is given as t^{\circ} C. To convert this to Kelvin, use: T = t + 273.
Total Power Radiated by the Sun: The Sun radiates energy over its entire surface area. Since the Sun can be approximated as a sphere with radius r, the total power P_{\text{total}} emitted by the Sun is: P_{\text{total}} = \sigma (t + 273)^4 \cdot 4 \pi r^2, where 4 \pi r^2 is the surface area of the Sun.
Intensity at Distance R: The power that reaches a unit area at distance R from the Sun is the power distributed over the surface area of a sphere with radius R. Therefore, the intensity I received is: I = \frac{P_{\text{total}}}{4 \pi R^2} = \frac{\sigma (t + 273)^4 \cdot 4 \pi r^2}{4 \pi R^2}.
Simplifying the expression gives: I = \frac{r^2 \sigma (t + 273)^4}{R^2}. This matches the first option, confirming that the correct power received by a unit surface is: r^2 \sigma (t + 273)^4 / R^2.

Thus, the correct answer is Option 1: r^2 \sigma (t + 273)^4 / R^2.

Assuming the sun to have a spherical outer surface of radius r, radiating like a black body at temperature $t^{\circ} C$, the power received by a unit surface, (normal to the incident rays) at distance $R$ from the centre of the Sun is:- Where $\sigma$ is the Stefan's Constant.

The Correct Option is A

Solution and Explanation

Top Questions on thermal properties of matter

Questions Asked in NEET (UG) exam