A-Level · Physics · AQA · Mark scheme decoded
AQA A-Level Physics: Stefan’s Law and Wien’s Displacement Law in Astrophysics — mark scheme explained
The short answer
Understanding the behavior of stars and other celestial bodies is a fundamental aspect of astrophysics. Two key laws that help us analyze these phenomena are Stefan's law and Wien's displacement law. These laws provide insights into the temperature, size, and power output of stars by examining their black-body radiation.
The question
A star has a surface area of 1.2 × 10 18 m 2 and a temperature of 6000 K. Calculate the power output using Stefan's law.
[Paraphrased for study — not reproduced from any exam paper.]
Mark scheme, decoded
How the examiner actually awards the marks on this topic.
Gradora's own decode of the marking approach — not the exam board's published mark scheme.
How marks are awarded
For numerical calculations, ensure that all steps are shown clearly. For explanations, use precise language and include relevant formulas where applicable. Always check units and significant figures in your final answer.
What the command words demand
- Calculate
- Perform a numerical calculation using the given data and appropriate formula.
- Estimate
- Make an approximate calculation, often using simplified assumptions or rounded values.
- Explain
- Provide a clear and concise explanation of a concept or process.
- Compare
- Identify similarities and differences between two or more concepts or objects.
Model answer
A full-mark response to the question above, worked through step by step.
Timing: Allocate about 5-7 minutes for each question involving numerical calculations and 3-4 minutes for questions requiring explanations or comparisons.
- Identify the values: A = 1.2 × 10 18 m 2 , T = 6000 K, σ = 5.67 × 10 -8 W m -2 K -4 .0 marks
- Use the formula P = AσT 4 :1 mark
- P = (1.2 × 10 18 ) × (5.67 × 10 -8 ) × (6000) 41 mark
- Calculate T 4 : (6000) 4 = 1.296 × 10 151 mark
- P = (1.2 × 10 18 ) × (5.67 × 10 -8 ) × (1.296 × 10 15 )1 mark
- P ≈ 8.8 × 10 25 W1 mark
Final answer: 8.8 × 10 25 W
Work through every step correctly and you earn all 5 marks.
Another worked example
A star emits the most radiation at a wavelength of 300 nm. Use Wien's displacement law to estimate the temperature of the star.
- Identify the values: λ max = 300 nm = 300 × 10 -9 m, constant = 2.9 × 10 -3 m K.1 mark
- Use the formula λ max T = constant:1 mark
- T = (2.9 × 10 -3 m K) / (300 × 10 -9 m)1 mark
- Calculate T: T ≈ 9670 K1 mark
Final answer: 9670 K
Work through every step correctly and you earn all 4 marks.
Common mistakes
Forgetting to convert units when using Wien's displacement law.
Why it happens: Students often forget to convert the wavelength from nanometers (nm) to meters (m) before applying the formula, leading to incorrect temperature calculations.
Fix: Always ensure that all units are consistent. Convert wavelengths from nm to m by multiplying by 10 -9 .
Using the wrong value for the Stefan-Boltzmann constant σ.
Why it happens: Students may use an incorrect value for σ, leading to significant errors in their calculations of power output.
Fix: Memorize or look up the correct value of σ, which is 5.67 × 10 -8 W m -2 K -4 .
Confusing the inverse square law with Stefan's law.
Why it happens: Students may mix up the formulas for the inverse square law and Stefan's law, leading to incorrect calculations of intensity or power output.
Fix: Understand that the inverse square law (I ∝ 1 / r 2 ) deals with how intensity decreases with distance, while Stefan's law (P = AσT 4 ) deals with the total power output based on temperature and surface area.
Forgetting to raise the temperature to the fourth power in Stefan's law.
Why it happens: Students may forget to apply the exponent of 4 to the temperature, leading to incorrect power output calculations.
Fix: Always remember to raise the temperature to the fourth power when using Stefan's law: P = AσT 4 .
Using the wrong constant in Wien's displacement law.
Why it happens: Students may use an incorrect value for the constant in Wien's displacement law, leading to incorrect temperature calculations.
Fix: Memorize or look up the correct value of the constant, which is 2.9 × 10 -3 m K.
Assuming that all stars are perfect black bodies without considering other factors.
Why it happens: Students may apply the laws of black-body radiation to stars without considering that real stars have variations in temperature and other complexities.
Fix: Understand that while treating stars as black bodies simplifies calculations, it is an approximation. Real stars can have variations in temperature and other factors that affect their radiation.
Where the marks go
The question types you’ll meet on this topic and the marks each one carries — so you know what to expect and where to focus.
| Question type | What you’re asked to do | Marks |
|---|---|---|
| Apply Stefan's Law | Calculate a star's power output from its surface area and temperature using Stefan's law. | 5 |
| Wien's Law Calculation | Estimate a star's temperature from its peak emission wavelength using Wien's displacement law. | 4 |
| Stefan's Law Calculation | Use Stefan's law with the star's radius and temperature to find its total power output. | 6 |
| Apply Wien's Law | Rearrange Wien's displacement law to calculate a star's temperature from its peak wavelength. | 4 |
| Total across these question types | 19 | |
Question types and mark tariffs are Gradora’s guidance based on how this topic is typically examined — not the board’s official paper structure.