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Theme B — The Particulate Nature of Matter

B.2.5 Energy Balance Problems — IB Physics HL Revision Notes & Topical Questions

5 min read 10 parts Lesson 23 of 53 1 quiz

📚 IB Physics HL Hub / Topical Revision
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Lesson summary

B.2.5 Energy Balance Problems — IB PHYSICS SL notes: Set up and solve energy-balance equations to find equilibrium temperatures of planets and predict the effect of changes in albedo, emissivity or solar intensity.Energy balance — if the incoming and outgoing energy are in balanc…

  • Self-correction
  • Intercepted power
  • Absorbed power
  • Per unit area of the planet
  • Solar constant at a planet
  • Radiating area

Part 1 of 8The principle of equilibrium

Energy balance
  • Energy balance if the incoming and outgoing energy are in balance, the Earth's temperature remains constant.
Condition
  • Condition the average power absorbed equals the average power radiated to space.
Warming
  • Warming if absorbed power exceeds emitted power, the planet warms.
Cooling
  • Cooling if emitted power exceeds absorbed power, the planet cools.
Self-correction
  • Self-correction as it warms, the emitted power rises as T4 until balance is restored.
Use
  • Use models like this let climate scientists predict temperature changes from greenhouse gas concentrations.
Method
  • Method every question is solved by writing power in = power out.

Part 2 of 8Power in

1Intercepted power
  • Intercepted power a planet of radius R receives S × πR2.
2Absorbed power
  • Pin = S(1 − a)πR2, where a is the albedo.
3Per unit area of the planet
  • (S/4)(1 − a).
4Solar constant at a planet
  • S = L/(4πd2), so it differs with distance.

Part 3 of 8Power out and equilibrium temperature

Radiating area
  • Radiating area the whole spherical surface 4πR2.
Outgoing power
  • Pout = eσ(4πR2)T4.
Setting equal
  • S(1 − a)/4 = eσT4, so T = [S(1 − a)/(4eσ)]1/4.
Radius cancels
  • T depends on distance (through S), albedo and emissivity.
No atmosphere
  • No atmosphere take e = 1.
With atmosphere
  • With atmosphere use an effective emissivity below 1, found as the intensity leaving the planet divided by σT4 for the surface temperature.

Part 4 of 8Two-body model of surface and atmosphere

Treatment
  • Treatment the surface and the atmosphere are treated as two black bodies at constant temperatures, with surface emissivity 1.
Surface receives
  • Surface receives the solar radiation that reaches it plus the radiation emitted by the atmosphere.
Atmosphere emits
  • I = eσTa4 with the atmosphere's emissivity e and temperature Ta.
Surface balance
  • Surface balance the total intensity absorbed by the surface equals σTs4, which gives the new surface temperature.
Change
  • Change subtract the old surface temperature to get the increase.

Part 5 of 8Sensitivity to changes

1Albedo increases
(more cloud or ice) — less power is absorbed, so T falls.
2Emissivity decreases
(more greenhouse gas) — the planet radiates less at a given T, so it warms until eσT4 again equals the absorbed power.
  • Since eT4 is constant, T ∝ e−1/4.
3Solar intensity increases
  • T ∝ S1/4, so a 10% rise in S raises T by only about 2.4%.
4Distance increases
  • S ∝ 1/d2, so T ∝ d−1/2.
5Fourth root
  • Fourth root temperatures react only mildly to large changes in input power.
6Small changes
  • ΔT ≈ ΔP/(4eσT3) per unit area.

Part 6 of 8Method for questions

1
Write the incoming intensity at the planet, using S or the inverse-square law.
2
Multiply by (1 − a) and divide by 4 for the average over the surface.
3
Equate with eσT4.
4
Rearrange for T in kelvin and take the fourth root carefully.
5
Judge whether the answer is realistic and, for Earth, compare with 288 K to infer the greenhouse effect.

Part 7 of 8Checking the answer

1Direction first
  • Direction first a more reflective planet has a smaller (1 − a), so T must be lower.
2Thicker atmosphere
  • Thicker atmosphere the effective emissivity shrinks, so T must be higher.
3Size of change
  • Size of change the fourth root keeps changes small, usually a few kelvin.
4State the model
  • "assuming a steady state, e = 1 and uniform temperature".
5Simplifications
  • Simplifications surface and atmosphere act as black bodies and stay at constant temperatures.

Part 8 of 8More from the specification

More from the specification
Planetary Energy Balance and Incident Radiation
  • A planet's global temperature remains stable when the total incident solar radiation absorbed by the planet equals the thermal infrared radiation radiated back into space by the Earth-atmosphere system.
Greenhouse Effect and Carbon Dioxide
  • Greenhouse gases (including carbon dioxide, water vapour,
  • and methane) absorb outgoing terrestrial infrared radiation emitted by the surface and re-radiate it in all directions, trapping thermal energy and elevating global temperatures.
Human Impact vs Stratospheric Ozone
  • Human industrial activities have substantially increased greenhouse gas emissions, driving anthropogenic global warming;
    • in contrast, stratospheric ozone functions primarily to absorb harmful incoming solar ultraviolet radiation rather than trapping thermal infrared radiation.
Infrared
  • The part of the electromagnetic spectrum emitted by warm objects.
Wavelength
  • Distance between a point on one wave and the same point on the next wave
Activity
  • Number of decays per second of a sample.
Infrared radiation
  • Long-wavelength thermal radiation emitted by the Earth

Key terms

TermMeaning
Energy balanceCondition in which average power absorbed equals average power radiated
Equilibrium temperatureSteady temperature at which power in equals power out
AlbedoFraction of incident radiation scattered by a planet
Effective emissivityEmissivity used to represent the greenhouse effect of an atmosphere
Solar constantIntensity of solar radiation at the planet's distance from the Sun
Radiative forcingChange in the energy balance due to a change in greenhouse gases

Avoid these mistakes

1
Using a temperature in °C in the Stefan–Boltzmann law — use kelvin.
2
Forgetting the factor of 4 between cross-section and surface area — divide S by four.
3
Forgetting (1 − a) — only the unreflected power is absorbed.
4
Getting the direction of a change wrong — more albedo cools, a thicker atmosphere warms.
5
Taking the fourth root incorrectly — check with an estimate.

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