Problem 118 - Master Resource Book in JEE Main Physics

Question

Assuming the sun to have a spherical outer surface of radius \(r\) radiating like a black body at temperature \(t^{\circ} \mathrm{C}\), the power received by a unit surface (normal to the incident rays) at a distance \(R\) from the centre of the sun is ( \(\sigma\) is stefan's constant) (a) \(4 \pi r^{2} \sigma t^{4}\) (b) \(\frac{r^{2} \sigma(t+273)^{4}}{4 \pi R^{2}}\) (c) \(\frac{16 \pi^{2} r^{2} \sigma t^{4}}{R^{2}}\) (d) \(\frac{r^{2} \sigma(1+273)^{4}}{R^{2}}\)

Step-by-Step Solution

Answer

The correct option is (b) \( \frac{r^2 \sigma (t+273)^4}{4\pi R^2} \).

Step 1: Convert Temperature to Kelvin

The temperature must be converted from Celsius to Kelvin because the Stefan-Boltzmann law uses Kelvin. The formula is: \[ T = t + 273 \] where \( t \) is the temperature in degrees Celsius and \( T \) is the temperature in Kelvin.

Step 2: Calculate Total Power Radiated by the Sun

The total power \( P \) radiated by the sun is given by the Stefan-Boltzmann law: \[ P = 4 \pi r^2 \sigma T^4 \] where \( r \) is the radius of the sun, \( \sigma \) is Stefan's constant, and \( T \) is the temperature in Kelvin.

Step 3: Power Received per Unit Area at Distance R

Since the power radiated spreads out uniformly over a spherical surface at distance \( R \), the power per unit area \( P_A \) at a distance \( R \) is given by:\[ P_A = \frac{P}{4 \pi R^2} = \frac{4 \pi r^2 \sigma T^4}{4 \pi R^2} \]Simplifying gives:\[ P_A = \frac{r^2 \sigma T^4}{R^2} \]Substitute \( T = t + 273 \):\[ P_A = \frac{r^2 \sigma (t+273)^4}{R^2} \]

Step 4: Determine the Correct Option

From the expression obtained in the last step, check with the given options:- Option (b) is \( \frac{r^2 \sigma (t+273)^4}{4\pi R^2} \).By comparing, option (b) is the one that correctly represents the power received by a unit surface area at a distance \( R \).

Key Concepts

Black Body Radiation

When a body emits radiation uniformly in all directions and absorbs all incident energy, it behaves as a black body. This is an idealized physical concept that helps us understand how objects emit energy. At any given temperature, a black body will radiate energy at a maximum possible rate for that temperature. The concept of black body radiation is important for understanding how the sun emits energy. By approximating the sun as a black body, we can use thermodynamic laws to calculate how much energy it emits. One of these laws is the Stefan-Boltzmann law, which says the total energy emitted by a black body is proportional to the fourth power of its temperature. This helps explain why even small increases in a black body’s temperature result in much greater energy emissions.

Temperature Conversion to Kelvin

The Kelvin scale is an absolute temperature scale used in science. It starts at absolute zero, the coldest possible temperature, which is -273.15 degrees Celsius. Unlike Celsius or Fahrenheit, Kelvin does not use degrees, making it the standard in scientific calculations involving temperature. When using the Stefan-Boltzmann law to calculate radiated power, temperatures must be in Kelvin. This is because the law depends on absolute temperatures rather than those relative to the freezing point of water. To convert Celsius to Kelvin, we simply add 273.15 to the Celsius temperature. In most scientific contexts, this simplifies to adding 273. This conversion ensures that the calculations remain consistent and the laws of thermodynamics are correctly applied.

Spheres and Power Distribution

Understanding how power spreads from a sphere is vital in astrophysics and related sciences. When the sun emits energy, it spreads uniformly across the surface of an expanding sphere. This is similar to how ripples spread out from a stone thrown into a pond. As the sphere's surface area grows with distance from the source, the power per unit area decreases, following an inverse square law. This means that the same amount of energy is distributed over a larger area as we move further away from the source, leading to less energy per unit area. For a spherical outer surface like the sun, the total power radiated is calculated using its radius and surface area. This leads us to use formulas involving the surface area of spheres to understand how much power any given area receives as energy radiates outward.

Power Per Unit Area Calculation

To understand how much of the sun's power reaches the Earth, or any point at a distance, we need to calculate power per unit area. This involves dividing the total power radiated by the surface area of the sphere at that distance.The calculation for power per unit area at a distance requires knowledge of the surface area of a sphere, which is given by \(4\pi R^2\). The formula also incorporates the total power calculated using the Stefan-Boltzmann law.By combining these calculations, we determine that power received per unit area decreases as the distance from the sun increases. For any specific case like our exercise, substitute the known values, such as temperatures in Kelvin and distances, into the formula to arrive at practical results.