
If an object at absolute temperature (T) radiates energy at rate $R$, then select the correct graph showing the variation of ${\log _0}R$ with ${\log _0}(T)$.
A)

B)

C)

D)

Answer
168.9k+ views
Hint: Rate of radiation of energy is known as power. We need to know the relation between rate of energy radiated and absolute temperature. Also understand what is meant by absolute temperature. Remember that absolute temperature and absolute zero temperature are related to each other but they are not the same.
Complete step by step solution:
When we start measuring temperature as $0K$ at the lowest possible energy state then temperature measured with respect to this zero is known as the absolute temperature of the object.
This measured $0K$ is known as the absolute zero temperature of the object.
Stefan-Boltzmann law gives us the relation between the power radiated and the absolute temperature of the object.
The law states that the power radiated by the object is directly proportional to the fourth power of the absolute temperature.
$P \propto {T^4}$
Mathematically, the equation is given as
$P = \varepsilon \sigma A{T^4}$
Where $P$ is the power radiated
$\varepsilon $ is the emissivity
$\sigma $ is the Stefan’s constant
$A$ is the radiating area
$T$ is the absolute temperature
Here in this question, the power radiated or the rate of energy radiated is represented as $R$ .
$ \Rightarrow R = \varepsilon \sigma A{T^4}$
Taking ${\log _0}$ on both sides of the equation, we get
$ \Rightarrow {\log _0}R = {\log _0}(\varepsilon \sigma A{T^4})$
$ \Rightarrow {\log _0}R = {\log _0}\varepsilon + {\log _0}\sigma + {\log _0}A + {\log _0}{T^4}$
$\because \varepsilon ,\sigma ,A$ are all constants ${\log _0}$ of all constants is also constant.
Therefore we can substitute ${\log _0}\varepsilon + {\log _0}\sigma + {\log _0}A = k$
$ \Rightarrow {\log _0}R = k + {\log _0}{T^4}$
We can represent this equation as a linear graph of ${\log _0}R$ VS ${\log _0}T$ with $k$ as the intercept.
Therefore, option $(A)$ is the correct graph where ${\log _0}R$ starts with an initial value of $k$ and then increases linearly with increase in the value of ${\log _0}T$ .
Note: Stefan’s constant $\sigma = 5.670 \times {10^8}watts/{m^2}{K^4}$ . It is the constant of proportionality in the Stefan-Boltzmann law. The emissivity $\varepsilon = 1$ for black body and $\varepsilon < 1$ for grey bodies. Stefan-Boltzmann law is used to calculate the temperature of the Sun.
Complete step by step solution:
When we start measuring temperature as $0K$ at the lowest possible energy state then temperature measured with respect to this zero is known as the absolute temperature of the object.
This measured $0K$ is known as the absolute zero temperature of the object.
Stefan-Boltzmann law gives us the relation between the power radiated and the absolute temperature of the object.
The law states that the power radiated by the object is directly proportional to the fourth power of the absolute temperature.
$P \propto {T^4}$
Mathematically, the equation is given as
$P = \varepsilon \sigma A{T^4}$
Where $P$ is the power radiated
$\varepsilon $ is the emissivity
$\sigma $ is the Stefan’s constant
$A$ is the radiating area
$T$ is the absolute temperature
Here in this question, the power radiated or the rate of energy radiated is represented as $R$ .
$ \Rightarrow R = \varepsilon \sigma A{T^4}$
Taking ${\log _0}$ on both sides of the equation, we get
$ \Rightarrow {\log _0}R = {\log _0}(\varepsilon \sigma A{T^4})$
$ \Rightarrow {\log _0}R = {\log _0}\varepsilon + {\log _0}\sigma + {\log _0}A + {\log _0}{T^4}$
$\because \varepsilon ,\sigma ,A$ are all constants ${\log _0}$ of all constants is also constant.
Therefore we can substitute ${\log _0}\varepsilon + {\log _0}\sigma + {\log _0}A = k$
$ \Rightarrow {\log _0}R = k + {\log _0}{T^4}$
We can represent this equation as a linear graph of ${\log _0}R$ VS ${\log _0}T$ with $k$ as the intercept.
Therefore, option $(A)$ is the correct graph where ${\log _0}R$ starts with an initial value of $k$ and then increases linearly with increase in the value of ${\log _0}T$ .
Note: Stefan’s constant $\sigma = 5.670 \times {10^8}watts/{m^2}{K^4}$ . It is the constant of proportionality in the Stefan-Boltzmann law. The emissivity $\varepsilon = 1$ for black body and $\varepsilon < 1$ for grey bodies. Stefan-Boltzmann law is used to calculate the temperature of the Sun.
Recently Updated Pages
Classification of Elements and Periodicity in Properties | Trends, Notes & FAQs

JEE Main 2021 July 25 Shift 1 Question Paper with Answer Key

JEE Main 2021 July 22 Shift 2 Question Paper with Answer Key

How to Calculate Moment of Inertia: Step-by-Step Guide & Formulas

Dimensions of Charge: Dimensional Formula, Derivation, SI Units & Examples

Dimensions of Pressure in Physics: Formula, Derivation & SI Unit

Trending doubts
JEE Main 2025 Session 2: Application Form (Out), Exam Dates (Released), Eligibility, & More

Uniform Acceleration

Displacement-Time Graph and Velocity-Time Graph for JEE

Atomic Structure - Electrons, Protons, Neutrons and Atomic Models

JEE Main 2025: Derivation of Equation of Trajectory in Physics

Instantaneous Velocity - Formula based Examples for JEE

Other Pages
NCERT Solutions for Class 11 Physics Chapter 1 Units and Measurements

JEE Advanced Marks vs Ranks 2025: Understanding Category-wise Qualifying Marks and Previous Year Cut-offs

NCERT Solutions for Class 11 Physics Chapter 2 Motion In A Straight Line

Units and Measurements Class 11 Notes: CBSE Physics Chapter 1

Motion in a Straight Line Class 11 Notes: CBSE Physics Chapter 2

Important Questions for CBSE Class 11 Physics Chapter 1 - Units and Measurement
