{ "format": "recallweave-deck/1", "title": "Coherent waves: from phasors to interference", "attribution": "Original questions, distractors, explanations and worked examples for RecallWeave, with AI assistance. Mathematical background: MIT MAS.160, Recitation 2 (Fall 2007), https://ocw.mit.edu/courses/mas-160-signals-systems-and-information-for-media-technology-fall-2007/66c9d296a29c7e96d7ceb3be945e2d71_rec2.pdf; MIT 8.03SC, Chapter 1 (Fall 2016), https://ocw.mit.edu/courses/8-03sc-physics-iii-vibrations-and-waves-fall-2016/8eb46d081c5ba4728cbbb369a03ba45e_MIT8_03SCF16_Text_Ch1.pdf; UNSW School of Physics, Physclips: Phasor addition, https://animations.physics.unsw.edu.au/jw/phasor-addition.html. This course uses a cosine convention and original numerical examples; no source passages, exercises or figures are copied.", "license": "Original question wording, explanations, worked examples and diagrams in this course: CC0-1.0. Referenced teaching materials retain their own licenses. No license change is made to other repository content.", "concepts": [ "complex-amplitude", "vector-addition", "relative-phase", "model-boundaries" ], "items": [ { "id": "phasor-real-part", "concept": "complex-amplitude", "prerequisites": [], "prompt": "Use the convention x(t) = Re{Z exp(i2πft)}, where f > 0 and Z = A exp(iφ), with A ≥ 0. Which real signal does Z represent?", "options": [ "A sin(2πft + φ)", "A cos(2πft − φ)", "A cos(2πft + φ)", "A sin(2πft − φ)" ], "answer": 2, "explanation": "Multiplying the exponentials gives A exp(i(2πft + φ)). Euler's formula makes its real part A cos(2πft + φ). Taking the imaginary part would give sine instead. The stated convention fixes both the projection and the phase sign.", "transfer": "For A = 2 and φ = π, write the real signal and evaluate it at t = 0." }, { "id": "phasor-quarter-cycle", "concept": "complex-amplitude", "prerequisites": [], "prompt": "A cosine-convention phasor starts at Z = 3 on the positive real axis. After a positive quarter-cycle rotation, what is the real signal's instantaneous value?", "options": [ "0", "3", "−3", "1.5" ], "answer": 0, "explanation": "A quarter cycle rotates 3 to 3i. The real signal is the horizontal, real-axis projection, which is zero at that instant. The arrow's length remains 3; instantaneous value and peak amplitude are different quantities.", "transfer": "Give the real-axis projections after one half-cycle and after one full cycle." }, { "id": "phasor-modulus", "concept": "complex-amplitude", "prerequisites": [], "prompt": "With x(t) = Re{Z exp(i2πft)}, a constant complex amplitude is Z = −3 + 4i. What is the real signal's peak amplitude?", "options": [ "1", "7", "−3", "5" ], "answer": 3, "explanation": "Peak amplitude is the nonnegative magnitude |Z| = sqrt((−3)² + 4²) = 5. Adding component magnitudes would give 7, and −3 is only the signal's value at t = 0. The imaginary component affects the phase as well as the magnitude.", "transfer": "For Z = 5 − 12i, calculate the peak amplitude and the initial real value separately." }, { "id": "phasor-negative-real", "concept": "complex-amplitude", "prerequisites": [], "prompt": "A nonzero cosine-convention phasor is Z = −2. Express phase in the interval [−180°, 180°). Which amplitude and phase describe it?", "options": [ "Amplitude −2, phase 0°", "Amplitude 2, phase −180°", "Amplitude 2, phase −90°", "Amplitude 2, phase 0°" ], "answer": 1, "explanation": "The magnitude is 2, and the arrow points along the negative real axis. Its phase is 180° modulo a full turn; the specified half-open interval represents that direction as −180°. A negative magnitude is not the amplitude convention used here.", "transfer": "What amplitude and principal phase describe Z = −2i in the same interval?" }, { "id": "phasor-in-phase", "concept": "vector-addition", "prerequisites": [ "complex-amplitude" ], "prompt": "Two real cosine signals have the same positive frequency, the same phase and common units. Their peak amplitudes are 1.5 and 2.5. What is the peak amplitude of their sum?", "options": [ "1", "4", "sqrt(8.5)", "3.75" ], "answer": 1, "explanation": "Equal phases point both phasors in the same direction. Their lengths add: 1.5 + 2.5 = 4. Subtracting applies to opposite phases, while sqrt(1.5² + 2.5²) would apply to a 90° relative phase.", "transfer": "Keep those amplitudes but turn the second phasor by 180°. Predict the new peak amplitude." }, { "id": "phasor-opposite-unequal", "concept": "vector-addition", "prerequisites": [ "complex-amplitude" ], "prompt": "At a common frequency, Z_A = 3 and Z_B = −1 in the same signal units. What constant resultant phasor represents the sum?", "options": [ "4", "−2", "2i", "2" ], "answer": 3, "explanation": "Complex amplitudes add component by component: 3 + (−1) = 2. Opposite directions reduce the resultant, but unequal magnitudes do not cancel completely. The positive real result has amplitude 2 and phase 0°.", "transfer": "Swap the magnitudes so Z_A = 1 and Z_B = −3. Give the resultant and its principal phase." }, { "id": "phasor-quadrature", "concept": "vector-addition", "prerequisites": [ "complex-amplitude" ], "prompt": "Two same-frequency cosine signals have constant phasors Z_A = 3 and Z_B = 4i. What is the peak amplitude of their combined real signal?", "options": [ "5", "7", "1", "sqrt(7)" ], "answer": 0, "explanation": "The resultant is 3 + 4i, so its magnitude is sqrt(3² + 4²) = 5. The individual peaks occur at different times because the relative phase is 90°. Adding peak magnitudes would predict 7 and overlook that timing difference.", "transfer": "Evaluate the two components and their sum at t = 0. Why is that sum smaller than the combined peak amplitude?" }, { "id": "phasor-resultant-phase", "concept": "vector-addition", "prerequisites": [ "complex-amplitude" ], "prompt": "Same-frequency component phasors add to Z = −2 − 2i. In the interval [−180°, 180°), what is the resultant's phase?", "options": [ "45°", "−45°", "−135°", "135°" ], "answer": 2, "explanation": "Both coordinates are negative, so the direction lies in the third quadrant. Equal coordinate magnitudes place it 45° past the negative real axis: 225°, represented as −135° in the requested interval. Using arctan(im/re) alone would lose the quadrant.", "transfer": "Find the principal phase of Z = −2 + 2i and explain which sign changed." }, { "id": "phasor-common-rotation", "concept": "relative-phase", "prerequisites": [ "complex-amplitude", "vector-addition" ], "prompt": "Two same-frequency phasors have fixed amplitudes and fixed relative phase. Rotate both by an additional 40° while keeping the time origin fixed. Which property of their sum must remain unchanged?", "options": [ "Its value at t = 0", "Its phase relative to the time origin", "The times of its peaks", "Its peak amplitude" ], "answer": 3, "explanation": "A common rotation rotates the vector sum by the same angle without changing its length. The resultant amplitude and cycle mean square stay the same. For a nonzero sum, its phase and peak times shift; its initial real projection can change.", "transfer": "Use the lab's 3-and-4 quadrature preset. Compare common phases 0° and 40° while keeping relative phase at 90°." }, { "id": "phasor-amplitude-law", "concept": "relative-phase", "prerequisites": [ "vector-addition" ], "prompt": "Two coherent cosine signals have peak amplitudes A and B, common units and relative phase δ. If R is their resultant peak amplitude, which expression is correct?", "options": [ "R² = A² + B² for every δ", "R² = A² + B² − 2AB cos δ", "R² = A² + B² + 2AB cos δ", "R² = (A + B)² for every δ" ], "answer": 2, "explanation": "Take one phasor along the real axis. The sum has coordinates A + B cos δ and B sin δ. Squaring and adding gives R² = A² + B² + 2AB cos δ. The relative phase controls the cross term.", "transfer": "Insert δ = 0°, 90° and 180°. Explain the three resulting amplitude rules." }, { "id": "phasor-complete-cancellation", "concept": "relative-phase", "prerequisites": [ "vector-addition" ], "prompt": "Both component amplitudes are strictly positive. Their real cosine signals have the same positive frequency and common units. What is required for their sum to vanish at every time?", "options": [ "Equal amplitudes and a 90° relative phase", "Equal amplitudes and a 180° relative phase modulo 360°", "Any amplitudes and a 180° relative phase modulo 360°", "Any amplitudes and an instant when both projections are zero" ], "answer": 1, "explanation": "An identically zero sum requires opposite, equal-length phasors. Equal amplitudes with a half-turn separation cancel at every instant. Opposite unequal phasors leave a nonzero difference, and a single zero crossing does not establish cancellation for an entire cycle.", "transfer": "In the cancellation preset, change only one amplitude from 2 to 2.1. Describe what remains." }, { "id": "phasor-cycle-mean-square", "concept": "relative-phase", "prerequisites": [ "vector-addition" ], "prompt": "Two same-frequency cosine signals each have peak amplitude 2 and a 60° relative phase. In squared signal units, what is the cycle mean square of their sum?", "options": [ "6", "4", "8", "12" ], "answer": 0, "explanation": "The resultant amplitude squared is 2² + 2² + 2×2×2 cos 60° = 12. A cosine's squared value averages to one half over a complete cycle, giving 12/2 = 6. The separate mean squares add to 4; the coherent cross term contributes the remaining 2. This is a mathematical signal average, not a calibrated intensity.", "transfer": "Keep both amplitudes at 2. Calculate the combined cycle mean square at 0° and at 180° relative phase." }, { "id": "phasor-different-frequencies", "concept": "model-boundaries", "prerequisites": [ "relative-phase" ], "prompt": "Two real cosines have frequencies 2 Hz and 3 Hz. Why does one constant resultant phasor rotating at a single shared frequency fail to represent their sum for all time?", "options": [ "Their relative phase changes with time.", "Their amplitudes must grow with time.", "Complex numbers cannot represent either component.", "Adding real signals requires equal instantaneous values." ], "answer": 0, "explanation": "The components rotate at different angular rates, so their relative phase is not constant. They can still be added at each time, but the fixed shared-frequency model in this lab does not apply. A momentary phasor picture cannot supply one constant resultant for the whole signal.", "transfer": "How many full turns of relative phase accumulate in one second between 2 Hz and 3 Hz?" }, { "id": "phasor-zero-phase", "concept": "model-boundaries", "prerequisites": [ "complex-amplitude", "relative-phase" ], "prompt": "Equal opposite phasors sum exactly to Z = 0. Which amplitude and phase should the lab report for that resultant?", "options": [ "Amplitude 0, phase 0° uniquely", "Amplitude 0, phase 180° uniquely", "Amplitude 0, phase undefined", "Amplitude undefined, phase 90°" ], "answer": 2, "explanation": "Zero has magnitude 0 but no direction. Any phase multiplied by zero describes the same zero signal, so no unique phase can be inferred. Reporting an arbitrary angle as a physical result would invent information. A small nonzero resultant still has a phase.", "transfer": "Explain why rounding a small residual to zero before calculating phase could hide a meaningful difference." }, { "id": "phasor-phase-lead", "concept": "model-boundaries", "prerequisites": [ "complex-amplitude" ], "prompt": "Under the stated cosine convention, compare y(t) = cos(4πt + π/2) with x(t) = cos(4πt), with t in seconds. What time shift expresses y relative to x?", "options": [ "y lags x by 0.125 seconds.", "y leads x by 0.125 seconds.", "y leads x by 0.25 seconds.", "y lags x by 0.25 seconds." ], "answer": 1, "explanation": "Write y(t) = cos(4π(t + 1/8)) = x(t + 0.125). It reaches corresponding phases one eighth of a second earlier, so it leads x. Positive phase in this convention is a time advance; the frequency is unchanged at 2 Hz.", "transfer": "Replace +π/2 with −π/2. State the corresponding lead or lag." }, { "id": "phasor-diagram-meaning", "concept": "model-boundaries", "prerequisites": [ "complex-amplitude", "vector-addition" ], "prompt": "The lab plots arrows on axes labeled Real and Imaginary for two scalar signals. What does an arrow's angle encode in this diagram?", "options": [ "The physical direction in which a wave travels.", "The location of a source in a room.", "The direction of energy transport in space.", "The signal's phase in the chosen cosine convention." ], "answer": 3, "explanation": "These axes hold complex-amplitude components. Arrow angle encodes phase; its real projection supplies the instantaneous scalar value after time rotation. The diagram contains no positions, travel directions, medium or energy-transport law. A physical application would need those additional definitions.", "transfer": "Name one extra physical assumption needed before interpreting a squared signal value as an intensity." } ] }