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77 lines
3.1 KiB
77 lines
3.1 KiB
id: phys-ex-conceptual-limits
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type: conceptual
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difficulty: medium
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points: 20
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related_lesson: phys-09
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question: |
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CONCEPTUAL UNDERSTANDING: Voltage vs Power Limitations
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A coiler claims: "I have 200 kW of power available in my DRSSTC, so I should
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easily get 10 m sparks!"
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Identify the flaws in this reasoning. Your answer should discuss:
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(a) Voltage vs power limitations
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(b) Energy per meter constraints
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(c) Capacitive divider effects
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(d) Realistic expectations
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hints:
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- "Consider both E-field requirements AND energy requirements"
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- "What happens to E_tip as spark grows?"
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- "How does capacitive division change with length?"
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- "Typical maximum spark lengths for Tesla coils"
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solution:
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answer: |
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FLAWS IN REASONING:
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(a) Voltage vs Power Limitations:
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Power alone doesn't determine spark length. The spark needs BOTH adequate
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electric field (E_tip > E_propagation ≈ 0.6 MV/m) AND sufficient energy.
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For a 10 m spark:
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- Average field needed: E_avg ≈ 0.6 MV/m (if κ ≈ 3)
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- This requires V_top ≈ 2,000 kV minimum
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- But typical Tesla coil voltages: 300-600 kV (factor of 3-7 too low!)
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- Voltage limitation dominates, not power
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(b) Energy Per Meter Constraints:
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Even if power is adequate:
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- For QCW with ε = 10 J/m: E_needed = 10 × 10 = 100 J
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- Time available: T ≈ 10-20 ms typical
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- Power needed: P = 100 J / 0.015 s = 6.7 kW (well below 200 kW!)
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- So power is not the limiting factor
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(c) Capacitive Divider Effects:
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As spark grows:
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- C_sh increases (≈ 6.6 pF/m, so 66 pF for 10 m)
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- V_tip = V_topload × C_mut/(C_mut + C_sh) decreases
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- For typical C_mut = 20 pF: V_tip = V_top × 20/86 = 0.23 × V_top
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- Lose 77% of voltage to division!
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- Combined with 1/L² field reduction: E_tip ∝ 1/L² catastrophic drop
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- This self-limiting effect prevents very long sparks
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(d) Realistic Expectations:
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- Burst mode record sparks: ~2-3 m
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- QCW mode record sparks: ~5-6 m
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- 10 m would require:
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* V_top ≈ 2+ MV (extreme)
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* Careful voltage ramping to fight division
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* Very large topload (high C_mut)
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* Sea level operation (higher E_propagation at altitude)
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- More power doesn't overcome voltage limit!
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- The claim confuses power-limited with voltage-limited regimes
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CORRECT REASONING:
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"I have adequate power, but am limited by achievable topload voltage and
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capacitive division effects. Realistic maximum is ~3-4 m for my coil,
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regardless of available power beyond ~20 kW."
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explanation: |
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This conceptual problem addresses a common misconception. Tesla coils are almost
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always voltage-limited, not power-limited. The E-field requirement (E_tip >
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E_propagation) combined with capacitive division creates a fundamental voltage
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barrier. Having excess power just makes the spark brighter and hotter, not longer.
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Understanding this distinction is critical for realistic performance expectations
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and efficient design decisions.
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related_concepts: ["voltage-vs-power", "limiting-factors", "capacitive-divider", "realistic-expectations"]
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