## ERQ · 12 marks · Topics: C.2 Wave model + D.2 Electric and magnetic fields
**Stem.** A student investigates microwaves using a horn transmitter T that emits a linearly polarised plane wave of frequency 10.0 GHz directed horizontally at a flat aluminium reflector R placed 0.500 m away. A small dipole receiver D, connected to a meter that reads a voltage proportional to the time-averaged power absorbed, is moved along the axis between T and R. A regular pattern of maxima and minima is observed. At the position of one maximum, with the dipole axis aligned parallel to the transmitter's electric field, the time-averaged intensity at D is measured to be 6.0 × 10⁻⁵ W m⁻². The distance between successive minima is found to be 1.55 cm.
### Part (a) State [2 marks] · AO1 · Topic: C.2
State **two** conditions necessary for the formation of the observed standing wave pattern between T and R.
### Part (b)(i) Calculate [2 marks] · AO2 · Topic: C.2
Using the student's measurement of the distance between successive minima, calculate the experimental value of the speed of the microwaves in air.
### Part (b)(ii) Determine [2 marks] · AO2/AO3 · Topic: C.2
Determine the percentage difference between the experimental speed found in (b)(i) and the accepted value c = 3.00 × 10⁸ m s⁻¹.
### Part (c) Show that [3 marks] · AO3 · Topic: C.2 + D.2
At a position of a standing-wave maximum the electric and magnetic field amplitudes of the resultant wave are E₀ and B₀ = E₀/c respectively, and the time-averaged intensity is given by the magnitude of the Poynting vector averaged in time, ⟨S⟩ = E₀B₀/(2μ₀). The dipole responds only to the component of **E** along its axis. Show that, when the dipole at this maximum is rotated by θ = 30° away from the field direction, the meter reading falls to approximately 4.5 × 10⁻⁵ W m⁻².
### Part (d) Discuss [3 marks] · AO3 · ASSUMPTIONS DISCRIMINATOR
The analysis in part (c) treats the field between T and R as a perfect standing wave formed by a single forward plane wave and its single reflection, with E and B in fixed perpendicular directions. Discuss **two** limitations of this idealisation, explaining in each case how the limitation would cause the measured intensity at the rotated dipole to differ from the value calculated in (c).
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## Mark Scheme
### Part (a) [2 marks]
- M1: Two coherent waves of the same frequency travelling in opposite directions along the axis (incident + reflected from R) [ECF: no]
- M2: Waves have (approximately) equal amplitude / are (approximately) parallel-polarised so that superposition produces nodes and antinodes [ECF: no]
### Part (b)(i) [2 marks]
- M1: Recognises distance between successive minima = λ/2, so λ = 2 × 1.55 cm = 3.10 × 10⁻² m [ECF: no]
- M2: v = fλ = (10.0 × 10⁹)(3.10 × 10⁻²) = 3.10 × 10⁸ m s⁻¹ (3 s.f., correct units) [ECF: yes]
### Part (b)(ii) [2 marks]
- M1: Correct method: |v_exp − c|/c × 100% [ECF: no]
- M2: (3.10 − 3.00)/3.00 × 100% ≈ 3.3% (accept 3–3.4%) [ECF: yes]
### Part (c) [3 marks]
- M1: Combines ⟨S⟩ = E₀B₀/(2μ₀) with B₀ = E₀/c to give ⟨S⟩ = E₀²/(2μ₀c); equivalently identifies I ∝ E₀² with both E and B contributing equally to the energy density (½ε₀E₀² = B₀²/(2μ₀)) [ECF: no]
- M2: Dipole detects only component E₀cosθ along its axis; meter reading ∝ (E₀cosθ)² because both E-field and associated B-field components projected onto the wave reduce by cosθ, so measured intensity = I₀cos²θ [ECF: yes]
- M3: I = (6.0 × 10⁻⁵)(cos30°)² = (6.0 × 10⁻⁵)(0.750) = 4.5 × 10⁻⁵ W m⁻² ✓ [ECF: yes]
### Part (d) [3 marks — award any 3 of the following, max 3]
- R1: Reflection at R is not perfect (aluminium has finite conductivity / edge diffraction), so the reflected amplitude < incident amplitude → standing wave is not pure; minima are non-zero and the cos²θ dependence sits on top of a travelling-wave background, so the rotated reading exceeds 4.5 × 10⁻⁵ W m⁻². [ECF: no]
- R2: The wave from the horn is not a true plane wave (it diverges / has curved wavefronts), so at off-axis or rotated orientations the field direction is not uniform across the dipole, weakening the strict cos²θ projection and lowering (or distorting) the reading. [ECF: no]
- R3: Multiple reflections (from the horn aperture, walls, the student) add stray fields whose polarisation direction differs from **E**₀, so the dipole picks up extra signal even at θ = 90°; the reading at 30° will not follow cos²θ exactly. [ECF: no]
- R4: The dipole has finite length comparable to λ (~3 cm) so it integrates field over a region where E varies (especially near a node–antinode structure); the assumption of a point-like projection onto a single E direction breaks down. [ECF: no]
- R5: Near the reflector the boundary condition forces E tangential → 0 but B tangential ≠ 0, so E and B are not simply in fixed perpendicular ratio B₀ = E₀/c at every point — the Poynting-vector argument for cos²θ assumes a free travelling wave, not the standing-wave field. [ECF: no]
### Marker notes
- (b)(i): Accept λ/2 stated in words or shown by diagram. Do not penalise units if SI throughout.
- (c) alternative: Candidate may write I = ½cε₀E₀² and (E_eff)² = (E₀cosθ)² directly; full credit provided the role of B₀ = E₀/c (or equivalent energy-density argument) is acknowledged for M1.
- (c): cos²30° = 3/4 must be used; arriving at 4.5 × 10⁻⁵ W m⁻² via I₀cosθ (= 5.2 × 10⁻⁵) earns M1 only.
- (d): Award at most 3 marks total; each reason must (i) name a specific failure of the idealisation and (ii) state the direction or nature of the resulting deviation. A reason without a stated consequence scores 0.