Physics • Year 12 • Module 7 • Lesson 2
Properties of Electromagnetic Waves
Build HSC Band 5–6 extended-response technique on inverse square law reasoning, wave interaction analysis, and evaluating real-world EM data.
1. Multi-step calculation — mobile phone tower safety (Band 5–6)
8 marks Band 5–6
Scenario. A 4G mobile phone tower in suburban Sydney emits 400 W of power uniformly in all directions. Australian Radiation Protection and Nuclear Safety Agency (ARPANSA) guidelines recommend that the general public should not be exposed to intensities above 10 W/m² for this frequency band. A resident’s house is 150 m from the base of the tower.
(a) Calculate the intensity of the radio waves at the resident’s house (150 m from the tower). 2 marks
(b) Calculate the minimum distance at which the intensity would equal the ARPANSA guideline value of 10 W/m². 2 marks
(c) A second tower is installed 300 m away from the same resident (double the distance of the first tower). Both towers emit 400 W. Determine the intensity at the resident’s house due to the second tower, and calculate the total intensity from both towers, stating any assumption you make. 2 marks
(d) State one limitation of the inverse square law model in this scenario and explain how it might cause the actual intensity at the house to differ from your calculated value. 2 marks
2. Data + scenario: solar constant and the inverse square law (Band 5–6)
8 marks Band 5–6
Scenario. The Sun emits approximately 3.85 × 1026 W of electromagnetic radiation in all directions. This value is called the solar luminosity. The table below shows measured solar intensities (called the “solar constant”) at various distances from the Sun.
| Planet / object | Distance from Sun (r, m) | Measured solar intensity (W/m²) | Predicted using I = P/(4πr²) (W/m²) |
|---|---|---|---|
| Mercury | 5.8 × 1010 | 9 130 | |
| Venus | 1.08 × 1011 | 2 620 | |
| Earth | 1.50 × 1011 | 1 361 | |
| Mars | 2.28 × 1011 | 590 | |
| Jupiter | 7.78 × 1011 | 50 |
Measured values adapted from NASA Planetary Fact Sheets (2024).
Q2. Analyse and evaluate the data in the table to determine whether the inverse square law accurately predicts solar intensities across the Solar System. In your response you must:
- Calculate the predicted intensity for at least three of the five planets/objects and compare with the measured values (show working).
- Assess whether the data support the inverse square law, using percentage agreement or discrepancy.
- Explain physically why intensity decreases with the square of distance from the Sun (reference spherical spreading of energy).
- State one reason why the measured and predicted values might not match exactly, with reference to physical processes in the Solar System or measurement.
- Evaluate the significance of the solar constant at Earth’s orbit (1361 W/m²) for life on Earth — what would change if Earth were at Mars’s distance?
3. Source critique — evaluate a student’s claim
6 marks Band 5–6
“I moved my Wi-Fi router from 2 metres away from my desk to 4 metres away. The signal dropped, but only by half — that’s because EM waves obey the inverse square law. Also, my microwave oven uses electromagnetic waves, so it must be dangerous to stand near it because EM waves keep on spreading out and penetrating everything in the room, just like X-rays from a hospital scanner.”
— Year 12 student, 2024.
3.1 Identify the scientific error in the student’s claim about the intensity drop from 2 m to 4 m. Calculate the correct factor by which intensity changes, and explain the inverse square law correctly. 2 marks
3.2 Identify and explain the error in the student’s comparison of microwave ovens with X-ray machines. In your answer, distinguish between the two types of radiation using frequency, energy, and ability to penetrate biological tissue. 2 marks
3.3 Explain whether the inverse square law applies to a Wi-Fi router inside a room, or whether it would give an overestimate or underestimate of intensity compared with a free-space point source. 2 marks
Q1(a) — Intensity at resident’s house (2 marks)
I = P/(4πr²) = 400/(4π × 150²) = 400/(4π × 22500) = 400/282743 ≈ 1.41 × 10−3 W/m² [1 mark for correct substitution; 1 mark for correct answer in correct units].
Note: This is well below the ARPANSA guideline of 10 W/m², so the resident is not at risk under this model.
Q1(b) — Minimum safe distance (2 marks)
Set I = 10 W/m²: r = √[P/(4πI)] = √[400/(4π × 10)] = √[400/125.66] = √3.183 ≈ 1.78 m. [1 mark for rearranging correctly; 1 mark for correct answer].
The intensity only reaches 10 W/m² within about 1.8 m of the tower base — far closer than any member of the public can typically approach.
Q1(c) — Second tower intensity and total (2 marks)
Assumption: the two waves are incoherent, so intensities add directly (no interference pattern). [required for full marks]
Second tower at 300 m: I⊂2; = 400/(4π × 300²) = 400/1130973 ≈ 3.54 × 10−4 W/m².
Check: I⊂2; = I⊂1;/4 = 1.41 × 10−3/4 = 3.53 × 10−4 W/m² — consistent. [1 mark]
Total: Itotal = 1.41 × 10−3 + 3.53 × 10−4 ≈ 1.76 × 10−3 W/m². [1 mark]
Q1(d) — Limitation of inverse square law model (2 marks)
The inverse square law assumes a point source radiating uniformly in all directions in free space. In suburban Sydney, buildings and terrain reflect radio waves, creating multipath propagation that can increase (or decrease) the local intensity compared with the free-space prediction. [1 mark for naming the limitation; 1 mark for explaining the physical cause and direction of effect].
Other acceptable answers: the antenna has a directional radiation pattern (gain in some directions); obstacles absorb or diffract the signal; the house walls and roof attenuate the signal, so indoor intensity is lower than the outdoor free-space value.
Q2 — Solar constant analysis (marking criteria, 8 marks)
Predicted values (using I = 3.85 × 1026/(4πr²)):
Mercury (5.8 × 1010): 3.85 × 1026/(4π × 3.364 × 1021) ≈ 9 100 W/m². Measured 9 130 → discrepancy <0.4%. ✓
Earth (1.50 × 1011): 3.85 × 1026/(4π × 2.25 × 1022) ≈ 1 362 W/m². Measured 1 361 → discrepancy <0.1%. ✓
Jupiter (7.78 × 1011): 3.85 × 1026/(4π × 6.05 × 1023) ≈ 50.6 W/m². Measured 50 → discrepancy <2%. ✓
Marking criteria (8 marks):
- [2] Correct calculation and comparison for at least three bodies, showing working (1 mark per correct pair of calculated + compared values, up to 2 marks for three pairs).
- [1] Assessment that data strongly support the inverse square law (<2% discrepancy for all five bodies), with percentage quoted or calculated.
- [1] Physical explanation: energy spreads over the surface area of an expanding sphere (4πr²); as r doubles, area quadruples, intensity quarters.
- [1] One valid reason for discrepancy: e.g., measurement is above the atmosphere (satellite-based, so accurate); if measured at surface, atmosphere absorbs & scatters some radiation, reducing measured value compared with model; solar wind particles carry some energy not in EM form; Sun is not a perfect point source (finite radius).
- [1] Comparison of Earth and Mars values: at Mars (r = 2.28 × 1011 m, I ≈ 590 W/m²), intensity is about 43% of Earth’s value. Less solar energy would mean lower average temperatures (well below 0 °C on most of Mars), reduced photosynthesis, and potentially no liquid water — life as we know it would be very difficult to sustain without significant adaptation or a thicker greenhouse atmosphere.
- [1] Explicit evaluative judgement: data strongly support the inverse square law for EM radiation across 8 × 1011 m scale with discrepancies under 2%.
Q3.1 — Error: intensity drop (2 marks)
The error is claiming intensity drops by half when distance doubles. By the inverse square law, intensity is proportional to 1/r². Moving from 2 m to 4 m doubles r, so the distance factor is (4/2)² = 4. Intensity drops to one-quarter (25%), not one-half. [1 mark for identifying the correct factor; 1 mark for explaining I ∝ 1/r²].
Q3.2 — Microwave vs X-ray error (2 marks)
The comparison is incorrect. Microwaves (frequency ~2.4 GHz) are non-ionising radiation; they heat tissue by causing water molecules to vibrate, but at microwave oven intensities outside the shielded cavity the leakage is minimal and non-ionising. X-rays (frequency ~1018 Hz) are ionising radiation with sufficient photon energy to eject electrons from atoms, damage DNA, and penetrate deep tissue. These are fundamentally different interactions. [1 mark for correctly distinguishing ionising vs non-ionising; 1 mark for frequency/energy contrast with reference to tissue penetration].
Q3.3 — Inverse square law in a room (2 marks)
The inverse square law gives an underestimate of intensity inside a room compared with free space. In a room, radio waves reflect off walls, floor, and ceiling and can add to the direct signal (multipath). Energy that would otherwise have spread further outward is “trapped” by reflections. Consequently, the actual intensity at the desk can be higher than the free-space prediction. [1 mark for stating the direction of discrepancy; 1 mark for explaining reflection/multipath trapping as the physical cause].