id: fund-ex-02c
type: multi-part
difficulty: medium
points: 15
related_lesson: fund-02
question: |
  A 4-foot spark is formed. The topload has C_topload = 30 pF when unloaded.

  (a) Estimate C_sh using the empirical rule
  (b) What is the total system capacitance with the spark?

  Hint: Consider how C_mut and C_sh combine in the circuit topology.

hints:
  - "Use C_sh ≈ 2 pF/foot for part (a)"
  - "The circuit is (C_mut || R) in series with C_sh"
  - "For DC or low frequency: R || C_mut looks like just C_mut"
  - "Series capacitors: C_total = (C₁ × C₂)/(C₁ + C₂)"

solution:
  steps:
    - "Part (a): Calculate C_sh = 2 pF/foot × 4 feet = 8 pF"
    - "Part (b): Recognize topload connects to (C_mut || R) in series with C_sh"
    - "At RF frequencies, parallel combination C_mut || R ≈ C_mut dominates"
    - "Series combination: C_series = (C_mut × C_sh)/(C_mut + C_sh)"
    - "Need to estimate C_mut. For typical geometries, C_mut ≈ 0.5 to 0.8 × C_topload"
    - "Assuming C_mut ≈ 0.6 × 30 pF = 18 pF (estimate)"
    - "C_series = (18 × 8)/(18 + 8) = 144/26 = 5.5 pF"
    - "Total system: C_total = C_topload + C_series (if in parallel to ground)"
    - "More accurately: Spark adds C_series in series, reducing total C seen from topload"

  answer_part_a: "8"
  unit_part_a: "pF"
  answer_part_b: "approximately 5.5 pF from spark circuit (but depends on C_mut estimate)"

explanation: |
  Part (a) is straightforward using the empirical rule. Part (b) is more complex
  because the spark adds a series combination of capacitances. The exact answer
  depends on C_mut, which must be determined from FEMM or estimated based on
  geometry. The key insight is that C_mut and C_sh form a capacitive divider that
  reduces the effective capacitance seen from the topload terminal.

  In practice, the "total system capacitance" includes the topload self-capacitance
  plus the series combination of the spark circuit elements. This problem illustrates
  why FEMM extraction is necessary for accurate modeling - C_mut cannot be calculated
  from simple formulas.

related_concepts: ["series-capacitance", "capacitive-divider", "total-capacitance", "FEMM-extraction"]