Stefan-Boltzmann Law


The radiation energy emitted per unit time by a body of surface area $A$, emissivity $e$, and absolute temperature $T$ is given by Stefan-Boltzmann Law \begin{align} \mathrm{d}E/\mathrm{d}t=e \sigma A T^4,\nonumber \end{align} where $\sigma=5.67\times10^{-8}\,\mathrm{W/m^2 K^4}$ is Stefan's constant. The emissivity of a body is defined as the ratio of radiation emitted by the body to the radiation emitted by a blackbody at the same temperature. Thus, emissivity of the blackbody is 1 and that of other bodies is between 0 and 1. The Stefan-Boltzmann can be derived from Planck's law by integrating the blackbody spectrum from wavelengths zero to infinity.

Stefan-Boltzmann Law
The radiation energy emitted per unit time per unit surface area by a blackbody depends on the fourth power of its absolute temperature i.e., $\mathrm{d}E/\mathrm{d}t=e \sigma A T^4$.

Solved Problems from IIT JEE

Problem from IIT JEE 2005

A body with area $A$ and emissivity $e=0.6$ is kept inside a spherical black body. Total heat radiated by the body at temperature $T$ is,

  1. $0.6\sigma eAT^4$
  2. $0.8\sigma eAT^4$
  3. $1.0\sigma eAT^4$
  4. $0.4\sigma eAT^4$

Solution: By Stefan-Boltzmann law, energy radiated per unit time by a body of surface area $A$, emissivity $e$, and absolute temperature $T$ is $e\sigma A T^4$.

Problem from IIT JEE 1994

Two bodies A and B have thermal emissivities of $0.01$ and $0.81$, respectively. The outer surface area of the two bodies are the same. The two bodies emit total radiant power at the same rate. The wavelength $\lambda_B$ corresponding to maximum spectral radiance in the radiation from B is shifted from the wavelength corresponding to maximum spectral radiance in the radiation from A by $1.00\;\mathrm{\mu m}$. If the temperature of A is 5802 K,

  1. the temperature of B is 1934 K.
  2. $\lambda_B=1.5\;\mathrm{\mu m}$.
  3. the temperature of B is 11604 K.
  4. the temperature of B is 2901 K.

Solution: Let $e_A=0.01$, $e_B=0.81$, and $T_A=5802 \mathrm{K}$. Stefan's law gives radiant power of two bodies as, \begin{align} {\mathrm{d}Q_A}/{\mathrm{d}t}=\sigma A e_A {T_A}^{\!4},\nonumber\\ {\mathrm{d}Q_B}/{\mathrm{d}t}=\sigma A e_B {T_B}^{\!4}.\nonumber \end{align} Equate ${\mathrm{d}Q_A}/{\mathrm{d}t}={\mathrm{d}Q_B}/{\mathrm{d}t}$ to get, \begin{align} T_B=\left({e_A}/{e_B}\right)^{\!1/4}T_A=1934\;\mathrm{K}.\nonumber \end{align} Wien's displacement law, $\lambda_m T=b$, gives, \begin{align} \lambda_B=(T_A/T_B)\lambda_A=3\lambda_A. \end{align} Also, since $\lambda_B > \lambda_A$ and $|\lambda_B-\lambda_A|=1\;\mathrm{\mu m}$, we get, \begin{align} \lambda_B-\lambda_A=1\;\mathrm{\mu m}. \end{align} Solve above equations to get $\lambda_A=0.5\;\mathrm{\mu m}$ and $\lambda_B=1.5\;\mathrm{\mu m}$. Thus, answer is A and B.

Problem from IIT JEE 2010

Two spherical bodies $A$ (radius 6 cm) and $B$ (radius 18 cm) are at temperatures $T_1$ and $T_2$, respectively. The maximum intensity in the emission spectrum of $A$ is at 500 nm and in that of $B$ is at 1500 nm. Considering them to be black bodies, what will be the ratio of the rate of total energy radiated by $A$ to that of $B$?

Solution: Stefan-Boltzmann law gives the energy radiated per unit time by a spherical black body of area $A=4\pi r^2$ and temperature $T$ as, \begin{align} E=\sigma A T^4=4\pi\sigma r^2 T^4. \end{align} The Wien's displacement law relates temperature of the black body to the wavelength at maximum intensity by \begin{align} \lambda_m T=b. \end{align} Eliminate $T$ from above equations to get, \begin{align} E=4\pi\sigma b^4 (r^2/\lambda^4),\nonumber \end{align} which gives, \begin{align} E_1/E_2=(r_1/r_2)^2\, (\lambda_2/\lambda_1)^4=9.\nonumber \end{align}

Questions on Stefan's Law

Question 1: Which of the following is the correct expression for Stefan's constant? (Here, h is Planck’s constant, c is the speed of light and k is Boltzmann’s constant)

A. $\frac{2\pi^5 c^2 h^3}{15 k^4}$
B. $\frac{2\pi^5 k^4}{15 c^2 h^3}$
C. $\frac{2\pi^5 c^4}{15 k^2 h^3}$
D. $\frac{2\pi^5 k^4}{15 c^3 h^2}$

Question 2: A blackbody radiates with a total intensity of $I = 5.68\, \mathrm{kW/m^2}$. At what wavelength does the spectral intesity $I(λ)$ peak?

A. $4.15 \,\mathrm{\mu m}$
B. $3.15 \,\mathrm{\mu m}$
C. $5.15 \,\mathrm{nm}$
D. $5.15 \,\mathrm{\mu m}$

Question 3: The figure shows intensity spectrum of a blackbody at two different temperatures $T_1=27$ deg C and $T_2=327$ deg C. If $A_1$ and $A_2$ are the areas under these curves then the ratio $A_1/A_2$ is

Problem on Stefan-Boltzmann Law
A. $1:8$
B. $\pi:16$
C. $1:16$
D. $\pi^2:16$

Related Topics

  1. Wien's Displacement Law
  2. Kirchhoff's Law of Radiation
  3. Newton's Law of Cooling

References and External Links

  1. IIT JEE Physics by Jitender Singh and Shraddhesh Chaturvedi
  2. Concepts of Physics Part 2 by HC Verma (Link to Amazon)
  3. Good Problems on Radiation - mostly related to astronomy
  4. Stefan's Boltzmann Law (Wikipedia)