University:
Santa Fe College
Course:
ETS 2527C | Electromechanical Components and Mechanisms
Academic year:

2024
Views:

169

Pages:

5
Author:

Jabin F.

Heading: Non-uniform Electric field: Equation: ⃗Fnet = 2(Δ⃗E) * ⃗p = 2(E2 - E1) * ⃗p Conclusion: ⃗Fnet ≠ 0 (Net force acting on an electric dipole in a non-uniform electric field is non-zero) Equation: ⃗F = -∇U = -∇(-⃗p⋅⃗E) Calculation: ⃗F = ∇(⃗p⋅⃗E) Conclusion: If the dipole is in the x-direction, then the electric field in the x-direction is responsible for the force. Heading: Torque: Equation: ⃗τ = ⃗p × ⃗E + (1/2) * ⃗p × (∇⃗E) Conclusion: The first term ⃗p × ⃗E is dominant. Note: If the dipole is like a point, the second term is negligible. Section 1 Diagram: Two dipoles (P1 and P2) separated by a distance r. Question: Calculate torque on P2 due to P1 and vice versa. Conditions: If dipole P2 is free to rotate in four dimensions, what direction will it point under equilibrium conditions? Question: Calculate the interaction energy of the system. Section 2 Solution (First case): τ2 = ⃗P2 × ⃗E1 ⃗E1 = (P1 / 4πεr^3) * (3cosθ * ⃗r - ⃗p1) ⃗E1 = (P1 / 4πεr^3) * ⃗r at θ = 90° τ2 = (P1 * P2) / (4πεr^3) * ⃗z Section 1 Equation: ⃗τ1 = ⃗p1 × ⃗E2 Calculation: ⃗E2 = (P2 / 4πε0r^3) * (2cosθ * ⃗r + sinθ * ⃗θ) Simplification: ⃗E2 = (P2 / 4πε0r^3) * (-2⃗r) Calculation: ⃗τ1 = (P1 * P2 * (-⃗z)) / (4πε0r^3) Section 2 Conclusion: P2 under equilibrium will be stable at parallel to the electric field ⃗E1. Equation: U = W = (1 / 4πε0r^3) * [⃗p1⋅⃗p2 - 3(⃗p1⋅⃗r)(⃗p2⋅⃗r)] Section 1 Conditions: θ = 90°, ⃗p1⋅⃗p2 = 0, ⃗p1⋅⃗r = 0 Conclusion: U = 0 Section 2 Calculation: ⃗p1⋅⃗p2 = |p1||p2| * cos(60°) = (p1 * p2) / 2 Calculation: ⃗p1⋅⃗r = |p1||r| * cos(30°) = (p1 * r) / 2 Substitution: U = (1 / 4πε0r^3) * [(p1 * p2) - 3 * (p1 * r)(p2 * r)] Simplification: U = (-3√3 * p1 * p2) / (16π * ε0 * r^3)