Classification by temperature, black-body radiation (A-level only) — AQA A-Level Physics

This topic covers how all objects emit thermal radiation across a continuous range of wavelengths, and how the spectrum depends only on the object's absolute temperature. You will study the idealised black body, which absorbs and emits all wavelengths perfectly, and learn the two key laws: Wien's displacement law, which relates peak wavelength to temperature, and the Stefan-Boltzmann law, which relates total power radiated per unit area to the fourth power of temperature.

Understanding black-body radiation is essential for astrophysics, climate science, and engineering. It explains why stars have different colours, why hot objects glow, and how the Earth's energy balance works. In AQA A-Level Physics, this topic is part of the thermal physics section and often appears in exam questions involving graph interpretation, calculations, and explanations of everyday phenomena.

• →A black body is a perfect absorber and emitter of all wavelengths of electromagnetic radiation; its spectrum depends only on its absolute temperature.
• →Wien's displacement law: lambda_max T = 2.9 x 10^-3 m K, where lambda_max is the peak wavelength and T is the absolute temperature in kelvin.
• →Stefan-Boltzmann law: the total power radiated per unit area is proportional to T^4, P/A = sigma T^4, where sigma = 5.67 x 10^-8 W m^-2 K^-4.
• →As temperature increases, the peak wavelength shifts to shorter wavelengths and the total intensity increases rapidly.
• →The area under a black-body radiation graph represents the total power radiated per unit area.

• States Stefan’s law as P = σAT⁴ or L = 4πr²σT⁴ and uses it to compare luminosities or radii of stars at different temperatures.
• States Wien’s displacement law as λmaxT = 2.9 × 10⁻³ m K and uses it to find peak wavelength or surface temperature.
• Explains that a black body absorbs and emits all wavelengths and that its spectrum depends only on temperature.
• Uses the two laws together to classify a star by temperature and to relate luminosity, radius and peak wavelength.
• Applies the inverse-square law for intensity where relevant, distinguishing luminosity from observed intensity at a distance.
• States that a black body absorbs all incident radiation and emits a continuous spectrum depending only on its absolute temperature.
• Describes the curve shape: intensity rises with wavelength to a single maximum, then decreases at longer wavelengths, with no crossing of curves for different temperatures.
• States Wien’s displacement law as λmaxT = 2.9 × 10⁻³ m K and identifies λmax as the wavelength of peak intensity.
• Converts a peak wavelength into metres and divides the constant by it to estimate T, or rearranges to find λmax from T.
• Explains that a higher temperature gives a shorter peak wavelength and a greater area under the curve, linking this to total emitted power.
• Uses correct units: λmax in m, T in K, and the constant in m K.
• States that intensity or flux is inversely proportional to the square of the distance from a point source, written as I ∝ 1/r² or b = L/(4πd²).
• Applies the law quantitatively, for example showing that moving from distance r to 2r reduces received intensity to one quarter of its original value.
• Identifies the point-source assumption: the source must be small compared with the distance at which it is observed, so that rays are effectively radial.
• Identifies the isotropic-emission assumption: power is radiated uniformly in all directions, so the same total power passes through every sphere of radius r.
• Explains that the intervening medium must be transparent, with no absorption or scattering, so that all emitted radiation reaches the observer.
• Uses the law to compare two situations by forming a ratio, such as b₁/b₂ = (d₂/d₁)², avoiding unnecessary substitution of constants.
• Recognises that the law is a limiting case and states a realistic condition under which it fails, such as observing a star from within its own planetary system or through dense interstellar dust.
• States Stefan's law as P = σAT⁴ and identifies each symbol, including T in kelvin and A as the emitting surface area.
• Uses A = 4πr² for a spherical star to obtain P = 4πσr²T⁴ and rearranges for radius, temperature or power as required.
• Forms a ratio for two stars so that σ and 4π cancel, for example P₁/P₂ = (r₁²T₁⁴)/(r₂²T₂⁴), and solves for the unknown quantity.
• Applies the fourth-power dependence correctly, for example recognising that doubling absolute temperature multiplies radiated power by sixteen at constant area.
• Compares stars of different temperature and size, explaining that a cooler star must have a larger radius to emit the same power as a hotter star.
• Converts temperatures from degrees Celsius to kelvin by adding 273 before substitution, and keeps consistent SI units throughout.
• States that the star is modelled as a black-body radiator, so the law gives the total power radiated across all wavelengths.

• 💡Quote the constant with its unit, 2.9 × 10⁻³ m K, and keep λmax in metres before converting to nm.
• 💡Show the rearrangement before substituting values, especially when finding T from λmax or r from L and T.
• 💡Use ratios when comparing two stars, since constants and 4π cancel and the arithmetic is simpler.
• 💡Sketch the axes with wavelength on the x-axis and intensity on the y-axis, label the peak and mark λmax clearly before substituting numbers.
• 💡Show the rearrangement of λmaxT = 2.9 × 10⁻³ m K explicitly, then substitute values with units so the kelvin unit follows through.
• 💡When comparing two sources, quote both peak wavelengths and state the inverse relationship: the shorter λmax corresponds to the higher temperature.
• 💡Write the relationship as a proportionality first, then convert to an equation with the 4πd² factor, so the geometry behind the law is visible.
• 💡When comparing two distances, use ratios and cancel constants; this reduces arithmetic and shows understanding of the inverse square behaviour.
• 💡Always name the assumptions explicitly when a question asks why a calculated value may be unreliable, linking each assumption to a physical situation.
• 💡Check units and significant figures at the end, and ensure distance is in metres before squaring.
• 💡Write the full equation P = 4πσr²T⁴ before substituting, then rearrange symbolically to reduce arithmetic errors.
• 💡For comparison questions, set up a ratio and cancel constants; this is quicker and less error-prone than calculating absolute values.
• 💡Check whether the question gives diameter or radius, and whether temperature is in kelvin, before starting the calculation.
• 💡Give the final answer with an appropriate unit and a sensible number of significant figures, and comment on whether the result is physically reasonable.
• 💡Always convert temperatures to kelvin before using any radiation law. Marks are often lost for using degrees Celsius.
• 💡When describing graphs, state the axes clearly, mention that the curve is continuous, and explain that the area under the curve equals total power per unit area.
• 💡For 'explain' questions, link the shape of the spectrum to the temperature: higher temperature means higher peak intensity and shorter peak wavelength.

• Using degrees Celsius in either law; correct this by converting to kelvin by adding 273.
• Forgetting to square the radius or to include 4π in L = 4πr²σT⁴; correct this by writing the full surface-area term.
• Assuming a cooler star is always less luminous; correct this by noting that luminosity also depends on radius, so a large cool star can outshine a small hot one.
• Using the wavelength in nanometres or micrometres directly in Wien’s law; the error is failing to convert to metres, so the correction is to multiply by 10⁻⁹ or 10⁻⁶ before dividing.
• Treating the peak wavelength as the maximum intensity value; the error is confusing the x-coordinate of the peak with the y-coordinate, so the correction is to read λmax from the wavelength axis.
• Assuming curves for different temperatures cross or that a cooler body can have a shorter peak wavelength; the correction is to remember that higher temperature always shifts the peak to shorter wavelength.
• Treating intensity as inversely proportional to distance rather than to distance squared; the correction is to write I ∝ 1/r² and check that doubling distance quarters the value.
• Applying the law to an extended source at close range without commenting on the point-source assumption; the correction is to state that the source must be far enough away that its angular size is negligible.
• Ignoring absorption or scattering by dust and gas and assuming all emitted power arrives; the correction is to state that the medium must be transparent for the inverse square law to apply directly.
• Confusing luminosity, a property of the source, with intensity or flux, which depends on distance; the correction is to define each symbol clearly before substituting.
• Substituting temperature in degrees Celsius instead of kelvin; the correction is to add 273 to convert to kelvin before raising to the fourth power.
• Using the diameter or circumference in place of the radius when calculating surface area; the correction is to use A = 4πr² with the radius, halving a diameter if necessary.
• Forgetting to square the radius when forming a ratio, or applying the fourth power to the radius; the correction is to keep r² and T⁴ separate in the ratio expression.
• Treating σ as a variable that must be evaluated when comparing two stars; the correction is to cancel it, along with 4π, in the ratio.
• Students often think a black body appears black at all temperatures. In fact, at high temperatures it glows, and at room temperature it appears black because it absorbs all visible light and emits mostly infrared.
• Many students confuse peak wavelength with maximum intensity, or think that a hotter object emits only shorter wavelengths. Actually, a hotter object emits more radiation at all wavelengths, but the peak shifts to shorter wavelengths.
• Students frequently forget to convert temperature to kelvin when using Wien's law or the Stefan-Boltzmann law, leading to incorrect answers.

1. 1Start by reviewing the electromagnetic spectrum and the difference between Celsius and kelvin. Practice converting temperatures.
2. 2Learn the definitions of a black body and the two key laws: Wien's displacement law and the Stefan-Boltzmann law. Write them down from memory and check units.
3. 3Sketch black-body radiation graphs for different temperatures, labelling axes and indicating how the peak and area change. Compare your sketches with textbook diagrams.
4. 4Work through calculation questions, including finding peak wavelength, temperature, or power ratios. Always show your working and units.
5. 5Attempt exam-style questions, especially those involving graph interpretation and explanations. Mark your answers using the mark scheme and note where you lost marks.

• 📋Calculation questions using Wien's law or the Stefan-Boltzmann law, often requiring temperature conversion and unit consistency. Advice: write down the formula, substitute values with units, and give your answer to an appropriate number of significant figures.
• 📋Graph interpretation questions where you must describe how the spectrum changes with temperature and explain the significance of the peak and area. Advice: use precise language such as 'peak wavelength shifts to shorter wavelengths' and 'area represents total power per unit area'.
• 📋Explanation questions about why a black body appears black at room temperature or why stars have different colours. Advice: link absorption and emission to temperature and the visible spectrum.
• 📋Comparison questions asking you to compare the radiation spectra of two objects at different temperatures. Advice: refer to both the peak wavelength and the total intensity, using the laws to support your answer.