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Thermal Physics
Stefan-Boltzmann Law of Thermal Radiation
Stefan's law states that the total radiant energy emitted per unit surface area of a black body across all wavelengths per unit time is directly proportional to the fourth power of its absolute temperature. The relation is expressed mathematically as E = sigma T^4.
UP TGT ScienceUP PGT PhysicsStudy Notes + Practice
Scope and prerequisites
TGT core
Definition of black body radiation, statement of Stefan's law, calculation of rate of heat emission and absorption, variation of temperature with time during cooling, and simple numericals based on E = sigma T^4 and Newton's law of cooling as a special case.
PGT extension
Thermodynamic derivation of Stefan-Boltzmann law using Maxwell's radiation pressure, connection with Planck's distribution law, Stefan-Boltzmann constant derivation, and advanced problems involving composite radiating surfaces and hollow enclosures.
Prerequisites
Concept of Temperature
Kinetic theory of gases
Law of equipartition of energy
Black Body Radiation and Emissive Power
A perfectly black body absorbs all radiations incident on it and reflects or transmits none, making its absorptivity equal to unity.
Total emissive power E represents the total thermal energy radiated per second per unit surface area of the body for all possible wavelengths.
The emissive power of a real body is less than that of a black body at the same temperature and is given by $E_real = e E_bb$, where e is the emissivity.
Statement and Mathematical Formulation of Stefan's Law
Formulated empirically by Josef Stefan in 1879 and derived theoretically by Ludwig Boltzmann in 1884 using thermodynamics.
The total energy emitted per second per unit area by an ideal black body is proportional to T^4, written as $E = \sigma T^4$.
If a black body is placed in an enclosure maintained at a lower absolute temperature T_0, the net rate of loss of heat energy per unit area is $E_net = \sigma (T^4 - T_0^4)$.
Stefan-Boltzmann Constant
The proportionality constant sigma is known as the Stefan-Boltzmann constant.
Its accepted SI value is approximately 5.67 \times 10^{-8} , W m^{-2} K^{-4}.
It fundamental links thermodynamics, electromagnetism, and quantum mechanics via radiation pressure and Planck's distribution.
Rate of Cooling of a Body
From Stefan's law, the rate of loss of heat dQ/dt for a body of surface area A and mass m with specific heat capacity s is given by mc(dT/dt) = -Asigma(T^4 - T_0^4).
For small temperature differences (T - T_0 ll T_0), Stefan's law reduces to Newton's law of cooling, showing linear dependence on temperature difference.
The time taken to cool from temperature T_1 to T_2 can be determined by integrating the differential cooling equation.
Derivations
Thermodynamic Derivation of Stefan's Law using Radiation Pressure
Conditions: Isotropic radiation field inside a hollow enclosure (cavity) of volume V at absolute temperature T.
Consider electromagnetic radiation inside a cavity as a photon gas exerting a radiation pressure $p = u / 3$, where u is the energy density of radiation per unit volume.
From the first and second laws of thermodynamics applied to a radiation chamber, the internal energy is U = u V and d$U = T dS - p dV$.
Using the Maxwell thermodynamic relation (partial S / partial V)_$T = (\partial p / \partial{T})_V$, substitute $p = u/3$ and d$U = V du + u dV$ to obtain the differential equation du / d$T = 4u / T$.
Integrating this differential equation yields ln(u) = 4 ln(T) + constant, which implies $u = a T^4$, where a is the radiation density constant.
Relating the energy density u to the total emissive power E of a black body through $E = (c/4)u results in E = (\frac{c a}{4}) T^4 = \sigma T^4$.
Result: $E = \sigma T^4$ where sigma = \frac{c a}{4}
Derivation of Newton's Law of Cooling from Stefan's Law
Conditions: A body at temperature T cooling in surroundings at constant lower temperature T_0, where (T - T_0) is very small.
Start with the net rate of heat loss from Stefan's law: \frac{dQ}{dt} = Asigma(T^4 - T_0^4).
Express the term (T^4 - T_0^4) as (T^2 - T_0^2)(T^2 + $T_0^2) = (T - T_0)(T + T_0)(T^2 + T_0^2)$.
Let $T = T_0 + \theta$, where \theta ll T_0. Then approximate (T + T_0) approx 2T_0 and (T^2 + T_0^2) approx 2T_0^2.
Substitute these approximations back to get \frac{dQ}{dt} = Asigma (T - T_0) (2$T_0)(2T_0^2) = 4Asigma T_0^3 (T - T_0)$.
Since \frac{dQ}{dt} = mc \frac{dT}{dt} = mc \frac{d\theta}{dt}, we obtain \frac{d\theta}{dt} = -K\theta, where $K = \frac{4Asigma T_0^3}{mc}$ is a constant.
Result: \frac{d\theta}{dt} = -K\theta
Formula sheet
Formula / law
Equation
Variables and units
Conditions
Stefan's Law for Black Body Emissive Power
$$E = \sigma T^4$$
E = Total emissive power, sigm$a = Stefan-Boltzmann constant$, T = Absolute temperature W m^{-2}
Ideal black body
Total Radiated Power from a Black Body
$$P = A \sigma T^4$$
P = Total power radiated, A = Surface area, sigm$a = Stefan-Boltzmann constant$, T = Absolute temperature W
Uniform temperature surface
Net Rate of Radiation Heat Exchange
$$P_{net} = A \sigma (T^4 - T_0^4)$$
$P_{net} = Net heat loss per second$, A = Surface area, T = Temperature of body, $T_0 = Temperature of surroundings$ W
Surrounding enclosure at constant temperature T_0
Net Rate of Radiation for a Grey Body
$$P_{net} = e A \sigma (T^4 - T_0^4)$$
e = Emissivity of the surface (0 < e le 1), A = Area, T = Body temperature, $T_0 = Surrounding temperature$ W
m = Mass of the body, c = Specific heat capacity, A = Surface area K s^{-1}
No heat loss by conduction or convection
Stefan-Boltzmann Constant Value
$$\sigma = 5.67 \times 10^{-8}$$
$\sigma = Stefan-Boltzmann constant$ W m^{-2} K^{-4}
Standard SI system
Energy Density of Radiation in Enclosure
$$u = a T^4$$
u = Radiation energy density, $a = Radiation constant (= 4sigma/c)$ J m^{-3}
Thermal equilibrium inside cavity
Radiation Pressure
$$p = \frac{u}{3} = \frac{4 \sigma T^4}{3 c}$$
p = Radiation pressure, c = Speed of light in vacuum N m^{-2}
Diffuse isotropic radiation field
Graphs and experiments
Emissive Power versus Temperature (E-T Graph)
Draw: A steeply rising non-linear curve plotting E on the y-axis against T on the x-axis, showing parabolic/hyperbolic steepness representing T^4 dependence.
Interpret: Demonstrates that radiant energy emission increases extremely rapidly at higher absolute temperatures.
Exam use: Frequently tested in multiple-choice questions asking to identify the correct power curve or ratio of emission at two temperatures.
Logarithmic Emissive Power Graph (log E vs log T)
Draw: A straight line plot of ln(E) versus ln(T) having a positive slope equal to 4 and y-intercept equal to ln(sigma).
Interpret: The straight line experimentally verifies the power-law relation E propto T^4 and allows direct graphical determination of Stefan's constant.
Exam use: Used in PGT level examinations to test experimental verification techniques and data linearization.
Rate of Cooling Curve (Temperature vs Time)
Draw: An exponential decay curve showing temperature T asymptotically approaching the surrounding temperature T_0 over time t.
Interpret: The slope of the tangent at any point on the curve gives the instantaneous rate of cooling dT/dt.
Exam use: Asked in descriptive problems involving cooling time calculations and comparison between Newton's linear approximation and Stefan's exact cooling.
Solved numericals
TGT core
A spherical black body of radius 2 m is maintained at 300 K. Calculate the total energy radiated per minute by the body. (Take sigm$a = 5.67 \times 10^{-8} W m^{-2} K^{-4})$
Identify the given parameters: Radius r = 2 m, Temperature T = 300 K, time t = 60 s.
Calculate the surface area of the sphere: $A = 4 \pi r^2 = 4 \times \pi \times (2)^2 = 16pi $, m^2.
Apply Stefan's law for total power radiated: $P = A \sigma T^4 = 16pi \times (5.67 \times 10^{-8}) \times (300)^4$.
Calculate total energy $Q = P \times t = 16 \times 3.1416 \times 5.67 \times 10^{-8} \times 8.1 \times 10^9 \times 60$.
Perform multiplication: Q approx 1.37 \times 10^6 J.
Answer: 1.37 \times 10^6 J
PGT extension
Two solid black spheres of radii r_1 and r_2 are made of the same material and have identical densities. They are heated to the same initial absolute temperature T and allowed to cool in the same surroundings. Find the ratio of their initial rates of cooling.
The rate of temperature fall is given by \frac{dT}{dt} = \frac{A sigma (T^4 - T_0^4)}{m c}.
Express surface area $A = 4 \pi r^2$ and mass $m = Volume \times density = (\frac{4}{3} \pi r^3) \rho$.
Substitute A and m into the cooling equation: \frac{dT}{dt} = \frac{4 pi r^2 sigma (T^4 - T_0^4)}{(\frac{4}{3} pi r^3 \rho) c} = \frac{3 sigma (T^4 - T_0^4)}{r \rho c}.
Notice that for spheres of the same material, \rho, c, sigma, T, and T_0 are constant, meaning \frac{dT}{dt} propto \frac{1}{r}.
Therefore, the ratio of initial rates of cooling is (\frac{dT}{dt})_1 / (\frac{dT}{dt})_$2 = r_2 / r_1$.
Answer: r_2 / r_1
PGT extension
A black body at temperature 227 °C radiates heat at a rate of 20 cal s^{-1} cm^{-2}. At what temperature will the rate of heat radiation become 320 cal s^{-1} cm^{-2}?
Convert initial Celsius temperature to Kelvin: $T_1 = 227 + 273 = 500 K$.
Substitute the given values: 320 / $20 = (T_2 / 500)^4$.
Simplify the ratio: $16 = (T_2 / 500)^4$.
Take the fourth root on both sides: $2 = T_2 / 500$.
Solve for T_2: $T_2 = 2 \times 500 = 1000 K$.
Answer: 1000 K (or 727 °C)
Assumptions and limitations
Stefan's law strictly applies only to ideal black bodies (absorptivity = 1); real surfaces require an emissivity factor e < 1.
The temperature of the body and surroundings must be measured on the absolute (Kelvin) scale.
The formula assumes uniform temperature across the entire radiating surface.
Common misconceptions
Common error: Stefan's law states that total energy radiated is proportional to the Celsius temperature raised to the fourth power.
Correct: It is proportional to the fourth power of the absolute temperature measured in Kelvin.
Zero emission occurs at absolute zero (0 K), which corresponds to -273.15 °C, making Celsius temperatures invalid for power-law scaling.
Common error: Newton's law of cooling is a completely separate law independent of Stefan's law.
Correct: Newton's law of cooling is an analytical approximation of Stefan's law valid only when the temperature difference between the body and surroundings is very small.
Binomial expansion of (T^4 - T_0^4) for small \theta leads directly to linear temperature dependence.
Common error: Emissivity is a constant universal property for all substances.
Correct: Emissivity depends heavily on the nature of the surface, its polish, and the temperature of radiation.
Metals and non-metals have varying emissivities that change with wavelength and thermal state.
Quick revision
Total emissive power of a black body is given by $E = \sigma T^4 with \sigma = 5.67 \times 10^{-8} W m^{-2} K^{-4}$.
Stefan's Law for Black Body Emissive Power: $E = \sigma T^4$
Total Radiated Power from a Black Body: $P = A \sigma T^4$
Must remember
The total emissive power of an ideal black body is directly proportional to the fourth power of its absolute temperature: $E = \sigma T^4$.
The Stefan-Boltzmann constant $\sigma$ has an accepted SI value of approximately $5.67 \times 10^{-8} \text{ W m}^{-2} \text{ K}^{-4}$.
When a body radiates heat into an enclosure at absolute temperature $T_0$, the net rate of heat loss per unit area is given by $E_{net} = \sigma (T^4 - T_0^4)$.
Newton's law of cooling is a special case approximation of Stefan's law valid strictly for small temperature differences where $(T - T_0 \ll T_0)$.
Radiation pressure inside an isotropic thermal enclosure is given by $p = \frac{u}{3} = \frac{4\sigma T^4}{3c}$.
Emissivity $e$ for a real grey body modifies the black body radiation equation to $P_{net} = e A \sigma (T^4 - T_0^4)$ where $0 < e \le 1$.
Glossary
Black BodyA theoretical body that absorbs all electromagnetic radiation incident on it, reflecting or transmitting none, with absorptivity equal to unity.
Emissive Power ($E$)The total amount of thermal radiation energy emitted per second per unit surface area of a body across all possible wavelengths.
Emissivity ($e$)The ratio of the emissive power of a real body to that of an ideal black body at the same temperature, ranging between 0 and 1.
Stefan-Boltzmann Constant ($\sigma$)The fundamental physical proportionality constant linking thermodynamic temperature to radiant energy flux, equal to $5.67 \times 10^{-8} \text{ W m}^{-2} \text{ K}^{-4}$.
Radiation Pressure ($p$)The mechanical pressure exerted on a surface by electromagnetic radiation due to momentum transfer, expressed as $p = u/3$.
Grey BodyA surface whose emissivity is independent of wavelength and remains constant across the spectral distribution at a given temperature.
Stefan's Law vs. Newton's Law of Cooling
Feature
Stefan's Law
Newton's Law of Cooling
Mathematical Form
$E = \sigma (T^4 - T_0^4)$
$\frac{dT}{dt} = -K(T - T_0)$
Temperature Range
Valid for all temperature differences
Valid only for small temperature differences ($(T - T_0 \ll T_0)$)
Origin
Thermodynamic and empirical derivation via $T^4$ dependence
Linear approximation derived by binomial expansion of Stefan's law
Ideal Black Body vs. Real Grey Body
Parameter
Ideal Black Body
Real Grey Body
Absorptivity ($a$)
$a = 1$
$a < 1$
Emissivity ($e$)
$e = 1$
$0 < e < 1$
Radiated Power Equation
$P = A \sigma T^4$
$P = e A \sigma T^4$
Practice Quiz
Choose an option to reveal the explanation.
1. What is the correct mathematical formulation of Stefan's law for the total emissive power of an ideal black body at absolute temperature $T$?
Explanation: Stefan's law states that the total radiant energy emitted per second per unit area by an ideal black body is directly proportional to the fourth power of its absolute temperature, expressed as $E = \sigma T^4$.
2. What is the accepted SI value and unit for the Stefan-Boltzmann constant $\sigma$?
Explanation: The Stefan-Boltzmann constant is fundamental in radiation physics and its accepted value is approximately $5.67 \times 10^{-8} \text{ W m}^{-2} \text{ K}^{-4}$.
3. If the absolute temperature of an ideal black body is doubled, by what factor does its total rate of energy radiation increase?
Explanation: According to $E = \sigma T^4$, if temperature $T$ becomes $2T$, the new emissive power is proportional to $(2T)^4 = 16T^4$, representing a 16-fold increase.
4. A black body is placed in an enclosure maintained at a constant lower absolute temperature $T_0$. What is the net rate of heat loss per unit area?
Explanation: The body emits radiation at rate $\sigma T^4$ and absorbs ambient radiation from the enclosure at rate $\sigma T_0^4$, making the net loss per unit area $\sigma (T^4 - T_0^4)$.
5. Under what condition does Stefan's law reduce to Newton's law of cooling?
Explanation: Binomial expansion of the term $(T^4 - T_0^4)$ when $(T - T_0)$ is very small compared to $T_0$ yields a linear relationship matching Newton's law of cooling.
6. What is the emissivity $e$ of a perfectly white or highly polished reflecting surface?
Explanation: A perfectly reflecting body absorbs no radiation and emits no thermal radiation, giving an absorptivity and emissivity equal to $0$.
7. What is the relation between radiation energy density $u$ inside a cavity at temperature $T$ and the Stefan-Boltzmann constant $\sigma$, given the speed of light $c$?
Explanation: Thermodynamic derivation connects energy density $u$ and emissive power via $u = \frac{4\sigma T^4}{c}$, where $c$ is the speed of light.
8. How is radiation pressure $p$ related to the radiation energy density $u$ inside an isotropic thermal enclosure?
Explanation: For a diffuse isotropic radiation field in a cavity, statistical mechanics and electromagnetic theory establish that radiation pressure is exactly one-third of the energy density: $p = u/3 = \frac{4\sigma T^4}{3c}$.
9. A sphere of radius $r$ is maintained at temperature $T$. What is its total radiated power assuming it acts as an ideal black body?
Explanation: The total surface area of a sphere of radius $r$ is $A = 4\pi r^2$. Therefore, the total radiated power is $P = A \sigma T^4 = 4\pi r^2 \sigma T^4$.
10. What happens to the rate of cooling $\frac{dT}{dt}$ of a solid metallic body as it cools down towards room temperature in a fixed enclosure?
Explanation: As the body cools, $T$ approaches $T_0$, making $(T^4 - T_0^4)$ approach zero, which causes the rate of cooling to decrease and vanish at thermal equilibrium.
11. Which physical domains are fundamentally linked by the Stefan-Boltzmann constant $\sigma$?
Explanation: Through Planck's radiation law and thermodynamic cavity relations, the constant $\sigma$ bridges thermodynamics, electromagnetic wave theory, and quantum mechanics.
12. Two black bodies $A$ and $B$ are at temperatures $1000 \text{ K}$ and $2000 \text{ K}$ respectively. What is the ratio of their emissive powers $E_A / E_B$?
Explanation: The ratio is $\frac{E_A}{E_B} = \left(\frac{T_A}{T_B}\right)^4 = \left(\frac{1000}{2000}\right)^4 = \left(\frac{1}{2}\right)^4 = \frac{1}{16}$.
13. If a grey body has an emissivity of $e = 0.6$ and surface area $2 \text{ m}^2$ at temperature $500 \text{ K}$, what is its total radiated power?
Explanation: Using the grey body radiation formula $P = e A \sigma T^4$, substituting $e = 0.6$, $A = 2$, $\sigma = 5.67 \times 10^{-8}$, and $T = 500$ gives option A.
14. Why are Celsius temperatures invalid for direct substitution into Stefan's law $E = \sigma T^4$?
Explanation: A body at $0 ^\circ\text{C}$ ($273.15 \text{ K}$) still radiates significant thermal energy. Power-law scaling requires absolute temperature measured from absolute zero.
15. In the differential equation for cooling of a body of mass $m$, specific heat $c$, and surface area $A$, what expression represents $\frac{dT}{dt}$?
Explanation: Equating heat lost per second $mc \frac{dT}{dt}$ to net radiation loss $A \sigma (T^4 - T_0^4)$ gives $\frac{dT}{dt} = -\frac{A \sigma (T^4 - T_0^4)}{mc}$.
PYQ Patterns
These are pattern-based practice questions. Authentic year labels are only used when verified from a source.
1. If the temperature of a black body is increased by $50\%$, the percentage increase in the rate of heat radiation from the body is approximately:
Explanation: Initial radiation $E_1 \propto T^4$. New temperature $T_2 = 1.5 T$. New radiation $E_2 \propto (1.5T)^4 = 5.0625 T^4$. Percentage increase = $\frac{5.0625 - 1}{1} \times 100\% = 406.25\% - 100\% = 306.25\%$, or approximately $306\%.$
2. Two solid spheres of the same material and same surface finish have radii $1 \text{ cm}$ and $3 \text{ cm}$ respectively. If they are maintained at the same high temperature, the ratio of their radiated powers is:
Explanation: Radiated power $P = A \sigma T^4 = 4\pi r^2 \sigma T^4$. Since material, temperature, and surface finish are identical, $P \propto r^2$. Thus, $\frac{P_1}{P_2} = \left(\frac{1}{3}\right)^2 = \frac{1}{9}$.
3. A metal sphere cools from $60 ^\circ\text{C}$ to $50 ^\circ\text{C}$ in $10 \text{ minutes}$. If the room temperature is $25 ^\circ\text{C}$, approximately how long will it take to cool from $50 ^\circ\text{C}$ to $40 ^\circ\text{C}$?
Explanation: Using Newton's law of cooling approximation: $\frac{T_1 - T_2}{t} = K \left(\frac{T_1 + T_2}{2} - T_0\right)$. For first interval: $\frac{60-50}{10} = K(55 - 25) \implies 1 = 30K$. For second interval: $\frac{50-40}{t} = K(45 - 25) \implies \frac{10}{t} = 20K$. Dividing gives $\frac{t}{10} = \frac{30}{20} \implies t = 15 \text{ minutes}$.
4. Assuming the sun to be a spherical black body of radius $7 \times 10^8 \text{ m}$ maintained at temperature $6000 \text{ K}$, what is the total power radiated by the sun? (Take $\sigma = 5.67 \times 10^{-8} \text{ SI}$ and $\pi \approx \frac{22}{7}$)
5. A cylindrical body of length $l$, radius $r$, and emissivity $e$ is heated to temperature $T$ and placed in a vacuum enclosure at temperature $T_0$. The net rate of heat loss by radiation is:
Explanation: The total surface area of a cylinder including its two circular ends is $A = 2\pi r l + 2\pi r^2$. For an infinitely long or thin cylinder where end areas are neglected or fully specified, total curved surface is $2\pi rl$. Considering standard textbook conventions for total surface area of a finite cylinder: $A = 2\pi r(r + l)$.
6. The energy density $u$ of thermal radiation inside a black-body cavity at temperature $T = 1000 \text{ K}$ can be estimated using the radiation constant $a$. What is the dimension of $a$?
Explanation: Since energy density $u = a T^4$, the units of $u$ are $\text{J m}^{-3}$ or $\text{N m}^{-2} = [M L^{-1} T^{-2}]$. Dividing by $T^4$ gives the dimensions $[M L^{-1} T^{-2} K^{-4}]$.
Timed Topic Test
Attempt these questions under a ten-minute limit using a separate timer.
1. What is the primary thermodynamic justification for the $T^4$ temperature dependence in Stefan's law?
Explanation: Ludwig Boltzmann derived Stefan's empirical law by applying the second law of thermodynamics to Maxwell's electromagnetic radiation pressure $p = u/3$ within a cavity.
2. Two concentric spherical shells of radii $R_1$ and $R_2$ ($R_2 > R_1$) act as black bodies at temperatures $T_1$ and $T_2$. What is the net radiation heat exchange between them per unit time?
Explanation: The inner sphere of radius $R_1$ radiates to the surrounding inner surface of the outer shell. The net heat exchange is determined entirely by the area of the emitting body facing the enclosure, $4\pi R_1^2 \sigma (T_1^4 - T_2^4)$.
3. A solid copper cube and a hollow copper sphere of the exact same mass and initial temperature $T$ are allowed to cool in the same ambient environment. Which one cools faster initially?
Explanation: A hollow sphere has a larger surface area-to-mass ratio ($A/m$) compared to a solid cube of equal mass. Since cooling rate $\frac{dT}{dt} \propto \frac{A}{mc}$, the hollow sphere cools faster.
4. If the absolute temperature of a radiating body decreases from $T$ to $0.5 T$, by what fraction does its net rate of radiation loss change in an enclosure at $0 \text{ K}$?
Explanation: With surroundings at $0 \text{ K}$, net loss is $E \propto T^4$. When $T$ becomes $0.5 T$, emission becomes $(0.5)^4 T^4 = \frac{1}{16} T^4$.
5. What is the radiation constant $a$ (often used in energy density $u = aT^4$) in terms of the Stefan-Boltzmann constant $\sigma$ and speed of light $c$?
Explanation: The radiation density constant $a$ is related to Stefan-Boltzmann constant by $a = \frac{4\sigma}{c}$, derived from integrating Planck's radiation distribution law over all wavelengths.
6. A body cools from $50 ^\circ\text{C}$ to $40 ^\circ\text{C}$ in $5 \text{ minutes}$ in a room at $20 ^\circ\text{C}$. Using Newton's cooling approximation, what is the temperature of the body after another $5 \text{ minutes}$?
Explanation: Applying exponential decay of temperature difference in Newton's law: $\frac{T - 20}{50 - 20} = e^{-kt}$. For first 5 min, ratio is $\frac{20}{30} = \frac{2}{3}$. For next 5 min, the temperature $T'$ satisfies $\frac{T' - 20}{40 - 20} = \frac{2}{3} \implies T' - 20 = 20 \times \frac{2}{3} = 13.33 \implies T' \approx 33.3 ^\circ\text{C}$.
7. How does the wavelength corresponding to maximum spectral emissive power ($\lambda_m$) shift as the absolute temperature $T$ of a black body increases according to Wien's displacement law, complementing Stefan's law?
Explanation: Wien's displacement law states $\lambda_m T = \text{constant}$, meaning as temperature $T$ increases, peak emission $\lambda_m$ shifts toward shorter wavelengths.
8. What does an emissivity value of $e = 1$ signify for a surface?
Explanation: An emissivity of $e = 1$ (along with absorptivity $a = 1$) defines an ideal black body that emits the maximum possible thermal radiation at any given temperature.
9. A thin square metal plate of side $L$ is heated uniformly on both sides to temperature $T$. What is its total thermal radiation power in an enclosure at $T_0$?
Explanation: A flat square plate has two active radiating surfaces (front and back), each of area $L^2$. Total effective area is $2L^2$, yielding power $P = 2L^2 \sigma (T^4 - T_0^4)$.
10. In stellar astrophysics, if a star's radius is halved and its surface absolute temperature is doubled, how does its total bolometric luminosity change?