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44 lines
1.4 KiB
44 lines
1.4 KiB
id: phys-ex-01a
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type: calculation
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difficulty: easy
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points: 10
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related_lesson: phys-01
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question: |
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A 0.8 m spark has V_top = 280 kV and tip enhancement factor κ = 4.
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(a) Calculate E_tip
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(b) If E_propagation = 0.5 MV/m, can the spark grow further?
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hints:
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- "E_average = V_top / L"
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- "E_tip = κ × E_average"
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- "Growth continues when E_tip > E_propagation"
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solution:
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steps:
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- "Part (a): Calculate average field"
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- "E_average = V_top / L = 280×10³ V / 0.8 m = 350 kV/m = 0.35 MV/m"
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- "Calculate tip field"
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- "E_tip = κ × E_average = 4 × 0.35 = 1.4 MV/m"
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- "Part (b): Compare to threshold"
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- "E_tip = 1.4 MV/m"
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- "E_propagation = 0.5 MV/m"
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- "E_tip > E_propagation ✓"
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- "Yes, spark can grow further"
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- "Safety margin: 1.4 / 0.5 = 2.8× above threshold"
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answer_a: "1.4"
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unit_a: "MV/m"
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answer_b: "yes"
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margin: "2.8"
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tolerance: 5.0
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explanation: |
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The tip field (1.4 MV/m) significantly exceeds the propagation threshold
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(0.5 MV/m), with a comfortable 2.8× safety margin. This spark is not voltage-
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limited and can continue growing. The enhancement factor κ = 4 concentrates the
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average field (0.35 MV/m) at the tip, creating sufficient field strength for
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sustained ionization and growth. If this spark has adequate power, it can extend
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well beyond 0.8 m.
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related_concepts: ["electric-field", "tip-enhancement", "growth-criterion", "voltage-limited"]
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