Electric Field Center Analysis: 6-Wire High-Voltage Transmission Line

Electric Field Superposition:

$$\vec{E}(\vec{r},t) = \sum_{i=1}^{6} \frac{\lambda_i(t)}{2\pi\varepsilon_0} \frac{\hat{r}_i}{r_i}$$

Line Charge Density (Conducting Cylinder):

$$\lambda_i(t) = \frac{2\pi\varepsilon_0 V_i(t)}{\ln(2h_i/a)}$$

Time-Varying Voltage:

$$V_i(t) = V_{\text{peak}} \sin(\omega t + \phi_i + \theta_{\text{circuit}})$$

Physics Principles:

• Conducting Boundary: $\vec{E} = 0$ inside conductors (radius $a = 1.5$ cm)
• Non-conducting Ground: No image charges or ground plane effects
• Line Charge Model: Each conductor treated as line charge at height $h_i = 44$ m above ground
• Phase Configuration: Circuit 1 & 2 base phases: [0°, 120°, 240°] (identical)
• Phase Offset Effect: $\phi_{C2,i}(t) = \phi_{C1,i} + \theta_{\text{offset}}$ applied in real-time

Phase Relationship Physics:

• θ_offset = 0°: Circuits have identical phases → Constructive interference at center → Maximum field
• θ_offset = 180°: Circuit 2 phases become [180°, 300°, 60°] → Perfect opposition → Destructive interference → Field cancellation
• Physical Mechanism: Each Circuit 1 conductor has equal/opposite Circuit 2 conductor on opposite side of center
• Spatial Symmetry: At center point, Circuit 1 and Circuit 2 contributions are equidistant but phase-opposed when θ_offset = 180°

Distance Calculations:

• Red Arrow (Center): $(0, 44)$ m
• Blue Arrow (25m lateral): $(25, 44)$ m
• Green Arrow (Ground level):
    Position: $(\cos(35°) \times 30, 2) = (24.6, 2)$ m
    Distance from center: $\sqrt{(24.6-0)^2 + (2-44)^2} = \sqrt{24.6^2 + 42^2} = 48.7$ m