id: phys-ex-01a
type: calculation
difficulty: easy
points: 10
related_lesson: phys-01
question: |
  A 0.8 m spark has V_top = 280 kV and tip enhancement factor κ = 4.

  (a) Calculate E_tip
  (b) If E_propagation = 0.5 MV/m, can the spark grow further?

hints:
  - "E_average = V_top / L"
  - "E_tip = κ × E_average"
  - "Growth continues when E_tip > E_propagation"

solution:
  steps:
    - "Part (a): Calculate average field"
    - "E_average = V_top / L = 280×10³ V / 0.8 m = 350 kV/m = 0.35 MV/m"
    - "Calculate tip field"
    - "E_tip = κ × E_average = 4 × 0.35 = 1.4 MV/m"
    - "Part (b): Compare to threshold"
    - "E_tip = 1.4 MV/m"
    - "E_propagation = 0.5 MV/m"
    - "E_tip > E_propagation ✓"
    - "Yes, spark can grow further"
    - "Safety margin: 1.4 / 0.5 = 2.8× above threshold"

  answer_a: "1.4"
  unit_a: "MV/m"
  answer_b: "yes"
  margin: "2.8"
  tolerance: 5.0

explanation: |
  The tip field (1.4 MV/m) significantly exceeds the propagation threshold
  (0.5 MV/m), with a comfortable 2.8× safety margin. This spark is not voltage-
  limited and can continue growing. The enhancement factor κ = 4 concentrates the
  average field (0.35 MV/m) at the tip, creating sufficient field strength for
  sustained ionization and growth. If this spark has adequate power, it can extend
  well beyond 0.8 m.

related_concepts: ["electric-field", "tip-enhancement", "growth-criterion", "voltage-limited"]