# 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.

## 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

A. $1:8$
B. $\pi:16$
C. $1:16$
D. $\pi^2:16$