Total Induced Charge on Plane | Santa Fe College - Edubirdie

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

2024
Views:

441

Pages:

2
Author:

Jabin F.

Definition of induced charge: q' = ∫σ ds q' = ∫∫∫ -q dm dy / (π(α(2+l)^3/2)) Change to polar coordinates: r^2 = α^2 + y^2 ds = γ dr dϕ = 2πr dr Integral in polar coordinates: q' = ∫₀^∞ -q d (2πr dr) / (2π(r^2+l^2)^3/2) Integration and simplification: q' = -2d (1/√(r^2+q^2)) ∫₀^∞ q' = -qd (1/d) q' = -q Integral expression: q' = -2d (∫₀^R 1/(r^2+d^2) dr) Evaluation of integral: q' = -2d (1/(R^2+d^2) + 1/d) Simplified result: q' = -q + 2d / √(R^2+d^2)

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