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109 lines
4.1 KiB
109 lines
4.1 KiB
id: phys-ex-comprehensive
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type: design
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difficulty: hard
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points: 100
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related_lesson: phys-09
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question: |
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COMPREHENSIVE SPARK PHYSICS DESIGN CHALLENGE
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Design a QCW coil from scratch to achieve 3.5 m sparks.
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Given constraints:
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- Budget allows C_primary up to 1.0 μF
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- V_primary limited to 600 V (safety)
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- Topload options: 20 cm toroid (C_top ≈ 25 pF) or 35 cm toroid (C_top ≈ 45 pF)
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- Target ramp time: 10-15 ms
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- Sea level operation (E_propagation = 0.6 MV/m)
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Complete the following analysis:
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1. Energy calculation:
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- Choose ε for QCW mode
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- Calculate total energy required for 3.5 m
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- Verify achievable with C_primary and V_primary
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2. Voltage requirement:
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- Estimate C_mut for each topload (use C_mut ≈ 0.7 × C_top)
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- Calculate C_sh for 3.5 m spark
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- For each topload, calculate V_topload needed for E_tip = 0.7 MV/m at 3.5 m (κ = 3.0)
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- Include capacitive division effects
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3. Power analysis:
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- For T_ramp = 12 ms, calculate required average power
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- Estimate peak power (assume 1.5× average for QCW)
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- Check if reasonable for DRSSTC primary
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4. Thermal verification:
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- Estimate leader diameter (2-4 mm typical)
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- Calculate thermal time constant
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- Verify ramp time << thermal time
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5. Final recommendation:
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- Which topload should be used?
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- Is 3.5 m target achievable?
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- If not, what would you change?
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hints:
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- "Use ε ≈ 10-12 J/m for QCW mode"
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- "Remember capacitive divider: V_tip = V_topload × C_mut/(C_mut + C_sh)"
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- "E_tip = κ × V_tip / L must exceed E_propagation"
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- "Thermal time: τ = d²/(4α) with α = 2×10⁻⁵ m²/s"
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solution:
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energy_calculation:
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chosen_epsilon: "11 J/m (typical QCW)"
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total_energy: "E = ε × L = 11 × 3.5 = 38.5 J"
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primary_check: "E_primary = 0.5 × C × V² = 0.5 × 1.0×10⁻⁶ × 600² = 180 J"
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verdict: "38.5 J << 180 J available ✓ Energy adequate"
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voltage_requirement:
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small_toroid:
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C_top: "25 pF"
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C_mut_est: "17.5 pF"
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C_sh: "23.1 pF (6.6 pF/m × 3.5 m)"
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V_tip_needed: "V_tip = E_prop × L / κ = 0.7×10⁶ × 3.5 / 3.0 = 817 kV"
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V_topload_needed: "V_top = V_tip × (C_mut + C_sh) / C_mut = 817 × 40.6/17.5 = 1,896 kV"
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verdict: "Unrealistic voltage required ✗"
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large_toroid:
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C_top: "45 pF"
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C_mut_est: "31.5 pF"
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C_sh: "23.1 pF"
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V_tip_needed: "817 kV (same)"
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V_topload_needed: "V_top = 817 × 54.6/31.5 = 1,416 kV"
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verdict: "Still very high, challenging ✗"
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power_analysis:
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ramp_time: "12 ms"
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avg_power: "P = E/T = 38.5 J / 0.012 s = 3.2 kW"
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peak_power: "~5 kW (1.5× average)"
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verdict: "Power requirement is modest ✓"
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thermal_verification:
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leader_diameter: "3 mm (estimate)"
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thermal_constant: "τ = (0.003)² / (4 × 2×10⁻⁵) = 113 ms"
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comparison: "T_ramp (12 ms) < τ (113 ms), ratio = 0.11"
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verdict: "Leader stays hot during ramp ✓ QCW condition satisfied"
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final_recommendation: |
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Neither topload can achieve 3.5 m with realistic voltages due to capacitive
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division. To achieve 3.5 m:
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Option 1: Accept shorter sparks (~2-2.5 m achievable with large toroid)
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Option 2: Increase primary voltage capability (beyond 600 V safety limit)
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Option 3: Use active voltage ramping to counteract division
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Option 4: Add intermediate electrode to reduce effective spark length
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Recommended: Use 35 cm toroid, target 2.5 m realistic goal, accept that
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voltage limitation dominates. Energy and power are adequate, but voltage
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limit prevents reaching 3.5 m target.
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explanation: |
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This comprehensive design challenge demonstrates the interplay between energy,
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voltage, and power limitations. The analysis reveals that voltage (electric field
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requirement) is the primary constraint, not energy or power. Capacitive division
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significantly increases the required topload voltage. The larger toroid helps but
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doesn't fully solve the problem. This is typical for Tesla coils - voltage-limited
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rather than power-limited. Realistic design must balance these constraints.
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related_concepts: ["design-integration", "voltage-vs-power-limits", "capacitive-divider", "QCW-optimization"]
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