--- id: phys-03 title: "Energy Per Meter Concept" section: "Spark Growth Physics" difficulty: "intermediate" estimated_time: 40 prerequisites: ["phys-01", "phys-02"] objectives: - Understand the concept of energy per meter (ε) for spark growth - Apply the growth rate equation dL/dt = P/ε - Calculate total energy and average power for target spark length - Recognize the difference between theoretical minimum and practical ε values tags: ["energy-per-meter", "epsilon", "growth-rate", "power", "ionization"] --- # Energy Per Meter Concept Extending a spark requires energy. Surprisingly, the energy needed is approximately **constant per unit length**, regardless of how long the spark already is. This fundamental concept enables practical spark growth modeling. ## The Energy Per Meter Parameter (ε) **Definition:** ε (epsilon) is the energy required to extend a spark by one meter. ``` Energy to grow from L₁ to L₂: ΔE ≈ ε × (L₂ - L₁) [Joules] where ε has units [J/m] ``` **Key characteristics:** - Approximately constant for a given operating mode - Independent of current spark length (first-order approximation) - Depends strongly on operating regime (QCW vs burst) - Empirical parameter that must be calibrated per coil **Why is this useful?** - Simple relationship: energy scales linearly with length - Easy to calculate power requirements - Enables growth rate predictions - Separates voltage limit (field) from power limit (energy) ## What Does ε Include? The energy per meter is **NOT** just the ionization energy. It includes all energy processes: ### 1. Initial Ionization Breaking molecular bonds to create ions and free electrons: ``` E_ionize ≈ 15 eV per molecule ``` ### 2. Heating to Operating Temperature Raising channel temperature from ambient to 5,000-20,000 K: ``` E_thermal = m × c_p × ΔT ``` ### 3. Work Against Pressure Expanding the channel against atmospheric pressure: ``` E_expansion = P × ΔV ``` ### 4. Radiation Losses Emitted light, UV, infrared, and RF: ``` E_radiation = ∫ σ T⁴ dA dt (blackbody + line emission) ``` ### 5. Branching Losses Energy wasted in short branches that don't contribute to main channel: ``` E_branching = ε × L_branches (failed growth attempts) ``` ### 6. General Inefficiencies Non-productive heating, turbulence, and other losses: ``` E_losses = various mechanisms ``` **Result:** Practical ε is 20-300× larger than theoretical ionization minimum! ## Theoretical Minimum Energy Let's estimate the absolute minimum energy needed for ionization alone: **Given:** - Ionization energy per molecule: ~15 eV - Air density: n ≈ 2.5×10²⁵ molecules/m³ - Channel diameter: d = 1 mm (typical) - Length increment: ΔL = 1 m **Calculation:** ``` Volume of 1 m channel: V = π(d/2)² × L = π(0.5×10⁻³)² × 1 = 7.85×10⁻⁷ m³ Number of molecules: N = n × V = 2.5×10²⁵ × 7.85×10⁻⁷ = 1.96×10¹⁹ molecules Energy to ionize: E_min = N × 15 eV × (1.6×10⁻¹⁹ J/eV) = 1.96×10¹⁹ × 15 × 1.6×10⁻¹⁹ = 0.47 J/m Theoretical minimum: ε_theory ≈ 0.3-0.5 J/m ``` **Why is practical ε so much higher?** Compare to real values: - QCW: ε ≈ 5-15 J/m (10-30× theoretical) - Burst mode: ε ≈ 30-100 J/m (60-200× theoretical) The difference accounts for: - Heating to high temperature (major contribution) - Radiation losses (visible light alone is significant) - Expansion work (pushing air aside) - Branching inefficiency (many failed paths) - Re-ionization (especially in pulsed modes) ## The Growth Rate Equation When the field threshold is met (E_tip > E_propagation), the growth rate is determined by power: ``` dL/dt = P_stream / ε [m/s] where: P_stream = power delivered to spark [W] ε = energy per meter [J/m] ``` **Physical interpretation:** - More power → faster growth - Higher ε (inefficiency) → slower growth for same power - Linear relationship: double power → double growth rate **When growth stops:** ``` If E_tip < E_propagation: dL/dt = 0 (stalled) Cannot grow regardless of available power (voltage-limited condition) ``` ### Predicting Growth Time For constant power during ramp: ``` Growth rate: dL/dt = P_stream / ε Integrating: L(t) = (P_stream / ε) × t Time to reach target length: T = ε × L_target / P_stream ``` **More realistic scenario:** Power changes as spark grows (loading changes): ``` T = ∫₀^L_target (ε / P_stream(L)) dL Requires simulation or numerical integration ``` --- ## WORKED EXAMPLE 3.2: Energy Budget **Given:** - Target spark length: L = 2 m - Operating mode: QCW with ε = 10 J/m - Growth time: T = 12 ms **Find:** (a) Total energy required (b) Average power required (c) If 80 kW is available, what changes? ### Solution **Part (a): Total energy** ``` E_total = ε × L = 10 J/m × 2 m = 20 J ``` Remarkably modest! Only 20 J to create a 2 m spark. **Part (b): Average power** ``` P_avg = E_total / T = 20 J / 0.012 s = 1,667 W ≈ 1.7 kW ``` For 12 ms growth, need ~1.7 kW average power. **Part (c): With 80 kW available** Available power is 80 kW, but only need 1.7 kW! ``` Power ratio: 80 kW / 1.7 kW = 47× more than needed ``` **Option 1: Grow much faster** ``` T_min = E_total / P_available = 20 J / 80,000 W = 0.00025 s = 0.25 ms (burst-like growth) ``` **Option 2: Grow to longer length (in same 12 ms)** ``` L_max_power = P_available × T / ε = 80,000 W × 0.012 s / 10 J/m = 960 J / 10 J/m = 96 m (!!) ``` **Reality check:** 96 m is absurd! What limits this? **Voltage limit kicks in first:** - Cannot maintain E_tip > E_propagation for 96 m - Spark stalls at voltage-limited length - Typical: L_max ≈ 2-4 m for practical topload voltages **Key insight:** Tesla coils are almost always **voltage-limited**, not power-limited. Excess power goes into brightening, heating, and branching rather than length. --- ## WORKED EXAMPLE 3.3: Comparing Operating Modes **Given:** - Two coils both deliver P = 50 kW average - Coil A: QCW mode, ε_A = 8 J/m - Coil B: Burst mode, ε_B = 50 J/m - Both operate for T = 10 ms **Find:** Which produces longer sparks? ### Solution **Coil A (QCW):** ``` L_A = P × T / ε_A = 50,000 W × 0.010 s / 8 J/m = 500 J / 8 J/m = 62.5 m (voltage-limited in practice) ``` **Coil B (Burst):** ``` L_B = P × T / ε_B = 50,000 W × 0.010 s / 50 J/m = 500 J / 50 J/m = 10 m (still voltage-limited in practice) ``` **Comparison:** ``` Ratio: L_A / L_B = ε_B / ε_A = 50/8 = 6.25× QCW coil produces 6.25× longer sparks for same power! ``` **Practical reality:** - Both limited by voltage before reaching these lengths - But ratio still applies: QCW gives much better length efficiency - Coil A might reach 2.5 m while Coil B reaches 0.4 m - Burst mode wastes energy on brightness and branching **Why choose burst mode then?** - Spectacular brightness and branches (visual appeal) - Higher peak current (electromagnetic effects) - Simpler drive electronics - Better for musical/modulated output - Different aesthetic goals than pure length --- ## Power-Limited vs Voltage-Limited Understanding the interplay between power and voltage limits: ### Voltage-Limited Condition ``` E_tip < E_propagation - Field too weak at tip - Spark cannot extend - More power → brighter/hotter, not longer - Common for Tesla coils ``` ### Power-Limited Condition ``` E_tip > E_propagation, but P_stream insufficient - Field adequate but not enough energy - Spark grows slowly or stalls before reaching potential - More voltage doesn't help without more power - Less common for Tesla coils (usually have excess power) ``` ### Practical Implications **For most Tesla coils:** 1. Design for adequate voltage (large topload, high primary voltage) 2. Ensure sufficient power (but don't need enormous amounts) 3. Optimize ε by choosing appropriate operating mode 4. Accept that voltage limit dominates final length **Rule of thumb:** - If P × T / ε >> L_actual, you're voltage-limited - If P × T / ε ≈ L_actual, you might be power-limited - Most coils fall in first category (voltage-limited) --- ## Key Takeaways - **ε definition**: Energy per meter [J/m], approximately constant for a given mode - **Growth rate**: dL/dt = P/ε when field threshold is met - **Total energy**: E_total ≈ ε × L (linear scaling) - **Theoretical minimum**: ε_theory ≈ 0.3-0.5 J/m (ionization only) - **Practical values**: 10-300× higher than theoretical (includes heating, radiation, losses) - **Operating mode matters**: QCW has low ε (efficient), burst has high ε (inefficient) - **Voltage limit dominates**: Most Tesla coils have more than enough power, limited by voltage ## Practice {exercise:phys-ex-03} **Problem 1:** A burst-mode coil has ε = 60 J/m. To reach L = 1.5 m in a 200 μs pulse, what power is required? Is this realistic? **Problem 2:** A QCW coil delivers 30 kW average power for 15 ms with ε = 12 J/m. Calculate: (a) Total energy delivered (b) Maximum length if power-limited (c) If actual length is only 1.8 m, what does this tell you? **Problem 3:** Explain why practical ε is 50-100× larger than the theoretical ionization minimum. List at least three major energy sinks. --- **Next Lesson:** [Empirical ε Values](04-empirical-epsilon.md)