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

2024
Views:

198

Pages:

5
Author:

Jabin F.

dipole moment P = ∫ rρ dτ r = rcosθẑ + rsinθcosφφ̂ + rsinθsinφŷ q = ∫∫∫ ρ(r²) r²sinθ dθdφdr Pz = ∫∫∫ ρ(r²) r²sinθ(rsinθcosθ) r²sinθ dθdφdr Px = ∫∫∫ ρ(r²) r²sinθ(rsinθsinφ) r²sinθ dθdφdr Py = ∫∫∫ ρ(r²) r²sinθ(rcosθ) r²sinθ dθdφdr ∫₀²π cosφ dφ = 0 = ∫₀π sinφ dφ (from (ii) & (iii)) If Φ has no dependence of φ then Pz and Py always zero. Now, Pz = ∫₀ᵃ ∫₀π ∫₀²π ρ₀(a² - r²)cosθ(rsinθ)sinθ dθ dφ dr Pz = 2πρ₀ (a⁴/4 - a⁶/6) ∫₀π cos²θ sinθ dθ Pz = 2πρ₀ (a⁴/4 - a⁶/6) (2/3) Pdipole = πρ₀a⁶/9 ẑ and E dipole ∝ 1/r³ if cosθ = 1 θ = 0° then monopole is non zero if cosθ = 0 θ = π/2 so E ∝ 1/r² Image Analysis Visual Elements: Handwritten text: The majority of the image is filled with handwritten text, likely notes or lecture slides. Diagrams: Two diagrams are drawn in the top part of the image, illustrating a quadrapole and its electric field lines. Equations: Numerous equations are written throughout the image, primarily related to the calculation of electric fields. Text Transcription: Ques. 16 26 Ques.17 P-26 equatorial Plan E = P/4πε₀r³ (cosθr̂ + sinθθ̂) (I) figure at point A, θ = 0 E = P/4πε₀r³ at point B, θ = 180° E = -P/4πε₀r³ Now (fieldline) Quadrapole- V ∝ 1/r⁴ (d) Ques. 16 26 monopole moment φ = 0 dipole moment P = 0 quadrapole moment is a tensor of Rank two so it has 9 component (3²) φij = \begin{matrix} φxx & φxy & φxz \ φyx & φyy & φyz \ φzx & φzy & φzz \end{matrix} i, j = x, y, z φij = ∫ [3rᵢ'rⱼ' - (r')²δᵢⱼ] ρ dτ δᵢⱼ = \begin{cases} 1 & if i = j \ 0 & if i ≠ j \end{cases} φij = [3rᵢ'rⱼ' - (r')²δᵢⱼ] q φxx = Σ [3rₓ'rₓ' - (r')²] q = [3(1)(1) - (r')²] q [3(-1)(-1) - (r')²] (-q) [3(-1)(-1) - (r')²] (-q) [3(1)(1) - (r')²] (2q) [3(1)(1) - (r')²] q If charge of place cut = plane then at (β) φxx = φyy = φzz = 0 φxy = 3(rx)(ry) - (r')²δij = 3(1)(1)(1+2) + 3(-1)(+1)(1+2) + 3(-1)(+1)(1+2) 3(1)(-1)(-2) φxy = 12q = φyx = φyx φyz = 3(1)(0)(1+2) + ... = 0 = φzy = φzx = φxz Put = \begin{matrix} 0 & 12q & 0 \ 12q & 0 & 0 \ 0 & 0 & 0 \end{matrix}