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Theme B — The Particulate Nature of Matter

B.1.14 Stefan-Boltzmann Law — IB Physics HL Revision Notes & Topical Questions

5 min read 9 parts Lesson 16 of 53 1 quiz

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B.1.14 Stefan-Boltzmann Law — IB PHYSICS SL notes: Use L = σAT⁴ to relate the luminosity, surface area and surface temperature of a black body or star.Stefan–Boltzmann law — the total energy emitted by a black body per unit area per second is proportional to the fourth power of i…

  • What the symbols mean
  • Doubling T
  • 10% rise in T
  • Luminous stars
  • Emissivity e
  • Real objects

Part 1 of 7The law

Stefan–Boltzmann law
  • Stefan–Boltzmann law the total energy emitted by a black body per unit area per second is proportional to the fourth power of its absolute temperature.
Equation
  • P = σAT4.
    What the symbols mean: L = luminosity, the total power emitted (W) · σ = 5.67 × 10−8 W m−2 K−4 · r = radius of the star (m) · A = surface area (m2) · T = surface temperature (K)
  • P is the total power emitted at all wavelengths in W.
  • σ = 5.67 × 10−8 W m−2 K−4 is the Stefan–Boltzmann constant.
  • A is the surface area in m2.
  • T is the absolute temperature in K.
Stars
  • Stars they are approximated as black bodies, so P is the luminosity L.
Sphere
  • A = 4πR2, so L = 4πσR2T4, where R is the radius and T the surface temperature.

Part 2 of 7Consequences of the T⁴ dependence

1Doubling T
  • Doubling T the power increases by a factor of 24 = 16.
210% rise in T
  • L increases by about 46%, since 1.14 = 1.46.
3Doubling the radius at fixed T
  • L increases by a factor of 4.
4Luminous stars
  • Luminous stars a star is luminous by being hot or by being large;
    • red giants are cool but luminous because their radius is huge.

Part 3 of 7Real objects and net power

Emissivity e
  • Emissivity e the ratio of the power radiated by a surface to that of a black body of the same temperature and dimensions;
    • for a perfect black body e = 1.
Real objects
  • P = eσAT4.
Stars in the IB
  • Stars in the IB they are usually treated as black bodies with e = 1.
Net power
  • Net power an object at T in surroundings at Ts radiates net power σA(T4 − Ts4), which is zero in equilibrium.

Part 4 of 7Finding stellar radii

Finding stellar radii
Find the luminosity from the brightness and distance, L = 4πd2b.
Find the surface temperature from the peak wavelength using Wien's law.
Rearrange to give the radius, R = √(L/(4πσT4)).
Ratios
  • L1/L2 = (R1/R2)2(T1/T2)4.
Comparison with the Sun
  • L☉ = 3.83 × 1026 W, R☉ = 6.96 × 108 m and T☉ ≈ 5800 K.
Doubling T and halving R
  • Doubling T and halving R the luminosity changes by (1/2)2 × 24 = 4.

Part 5 of 7Equilibrium temperature of a planet

Absorbed
  • Absorbed a planet at distance d from a star of luminosity L absorbs (1 − α)(L/4πd2)πr2 over its cross-section, where α is the albedo.
Emitted
  • Emitted it emits σ(4πr2)T4 over its whole surface.
Result
  • Result setting absorbed equal to emitted gives T = [(1 − α)L/(16πσd2)]1/4, independent of the radius.
Earth
  • Earth with α ≈ 0.3 this gives about 255 K; the actual surface is warmer because of the greenhouse effect.

Part 6 of 7Using the law

Typical question
  • Typical question given three of L, A (or R) and T, find the fourth, or compare two stars by ratios.
Convert
  • Convert temperature to kelvin.
Surface area
  • Surface area use the surface area of the sphere, not its volume.
Powers of ten
  • 104 K becomes 1016 K4.
Dimension check
  • Dimension check include W m−2 K−4 for σ in the working.

Part 7 of 7More from the specification

1
Thermal radiant power from hot solids
  • Thermal radiant power from hot solids radiant energy emitted per square metre of incandescent solids scales proportionally with absolute temperature to the fourth power.
2
Stefan-Boltzmann Law of Blackbody Radiation
  • The total thermal power P emitted by a blackbody radiator is directly proportional to surface area A and the fourth power of thermodynamic temperature T: P = eσAT⁴.
3
Emissivity (e) of Real Surfaces
  • Emissivity e is a dimensionless parameter (0 ≤ e ≤ 1) comparing an object's radiant power to an ideal blackbody radiator;
    • polished copper has low emissivity,
  • while matte black has e ≈ 1.
4
Wien's Displacement Law
  • The peak emission wavelength λ_max of continuous blackbody spectra is inversely proportional to absolute temperature, shifting toward shorter wavelengths as temperature rises.
    • (λ_max * T = 2.90 × 10⁻³ m K)
5
Thermal Calorimetry with Liquid Water
  • In laboratory calorimetry, a hot solid copper block submerged into cold water reaches thermal equilibrium when heat dissipated by the metal equals heat absorbed by the liquid.
6
Solid
  • State with a fixed shape and a fixed volume.
7
Fusion
  • Joining of light nuclei into a heavier one.
8
Intensity
  • Power per unit area perpendicular to the wave direction.
9
Kinetic energy
  • K = ½mv2, energy of motion.
10
Apparent brightness
  • The power received per unit area at the observer
11
Phase change
  • Change of matter from one state to another
12
Thermal conductivity
  • Material constant relating the rate of conduction to the temperature gradient and area
13
Internal energy
  • Total random kinetic and potential energy of all particles.
14
Thermal
  • A convection current in the air.
15
Liquid
  • State with a fixed volume that takes the shape of its container.
16
Copper
  • Metal widely used for electrical wires.
17
Convection
  • Transfer of thermal energy by the movement of a heated fluid
18
Conduction
  • Transfer of thermal energy by lattice vibrations and free electrons
19
Molecule
  • Tiny particle of the fluid, too small to be seen.
20
Capacity
  • Charge a cell can supply, in A h or C.
21
Density
  • Mass per unit volume.

Key terms

TermMeaning
Stefan–Boltzmann lawPower radiated by a black body is proportional to its area and to T4
Stefan–Boltzmann constantσ = 5.67 × 10−8 W m−2 K−4
LuminosityTotal power radiated by a star
EmissivityRatio of the power radiated by a surface to that of a black body at the same temperature
Surface area of a sphere4πR2
Net radiated powerPower emitted minus power absorbed from the surroundings

Avoid these mistakes

1
Using °C in the law — use kelvin.
2
Using volume instead of surface area — use 4πR2.
3
Thinking doubling T doubles the power — it increases it sixteen-fold.
4
Forgetting emissivity for non-black bodies — P = eσAT4.
5
Forgetting the surroundings for net power — subtract Ts4.

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